6d24dc0
HARMONIC FUNCTIONS, CURVATURE, AND A COUNTING SURPRISE
A complete proof of a counterexample, written to be read front to back.
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READ ME FIRST
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This document proves one surprising inequality about curved spaces.
We build one specific space, we count how many independent "equilibrium"
functions of a certain size it supports, and we show the count is larger
than the count in ordinary flat space. The extra hypothesis is that the
space is not bending inward too much (nonnegative Ricci curvature). The
surprise is that this gentle hypothesis does not keep the count from
beating flat space.
Every term is defined the first time it appears. The mathematical tools
are built from scratch: the Laplacian, curvature, and the counting
argument are all explained. A few computations near the end are long but
elementary; they are shown in full because the whole point is to prove
the theorem, not to quote it. Symbols that might be unfamiliar are
collected in the GLOSSARY at the very end.
The whole result in one paragraph:
Let H_k(M,g) be the vector space of harmonic (Laplacian-zero) functions
on an n-dimensional curved space (M,g) whose growth is at most degree k,
in the sense that |u(x)| is bounded by a constant times (1 + distance from
a base point)^k at every point x simultaneously. Let h_k be its
dimension. In flat Euclidean space R^n this count is a known binomial
number. We construct a complete smooth metric g on R^n, with nonnegative
Ricci curvature and Euclidean distance from the origin, for which
h_k(R^n,g) is strictly larger than the flat-space count, for some even
n at least 8 and some integer k at least 2. We can even arrange
n = 16 and k = 50000 with an explicit integer computation. The metric is
Euclidean near the origin, has asymptotic volume ratio strictly between
zero and one, and has more than one tangent cone at infinity. This
answers the integer-degree form of Yau's dimension comparison question
in the negative.
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HOW TO READ THIS
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Parts 1 and 2 state the question and the answer, and explain the strategy
in plain words. Parts 3 through 9 are the proof: they build the
ingredients one at a time. Part 10 records an explicit finite arithmetic
selection that pins down concrete numbers n = 16 and k = 50000. If you
only want the story, read Parts 1, 2, and the statement of the main
result at the top of Part 9, then stop. If you want the whole proof,
read everything in order.
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PART 1. THE QUESTION
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1.1 Curved spaces and their metrics.
A RIEMANNIAN MANIFOLD (M^n, g) is an n-dimensional smooth space M
together with a METRIC g, which is a rule that assigns to each point a
way to measure lengths of tangent vectors and hence lengths of paths,
angles, and volumes. The two basic examples are:
- flat n-dimensional space R^n with the ordinary Euclidean metric,
which we call g_E;
- the round sphere S^m of radius one, the surface of a ball in
(m+1)-dimensional space.
A COMPLETE space is one in which every path that ought to converge
actually does, in the metric sense: equivalently no path can run off to
a place that is not part of the space within finite length. R^n with the
Euclidean metric is complete. The spheres we build our links from,
being compact, are complete.
The DISTANCE d_g(x,y) is the length of the shortest path from x to y
joining them inside M. Completeness of the metric makes such shortest
paths exist between nearby points and makes distance balls compact.
1.2 The Laplacian.
On R^n the Laplacian of a smooth function is
Laplacian(u) = d^2u/dx_1^2 + ... + d^2u/dx_n^2.
On a curved space it is defined by taking the same sum of second
derivatives but weighted by the way the space bends; there is a unique
operator without any extra terms that is invariant under the symmetries
of the metric. We only need three facts about it in this document:
(i) On flat R^n it is the ordinary sum of second derivatives.
(ii) On the round unit sphere it has eigenvalue B_l = l(l+m-1) on the
space of restrictions of homogeneous harmonic polynomials of
degree l in m+1 variables. These are the SPHERICAL HARMONICS
V_l.
(iii) The Laplacian is the rate of change of a particular "energy";
in particular, its sign tells us whether a function wants to
smooth out (Laplacian negative) or oscillate (Laplacian
positive).
1.3 Harmonic and polynomial-growth functions.
A function u on (M,g) is HARMONIC if
Laplacian(u) = 0.
Equivalently (mean-value property) the value u(x) equals the average of
u over tiny spheres centered at x. In physics terms harmonic functions
are equilibrium solutions: they do not change on average as one moves
around.
Fix a base point o. Say u has POLYNOMIAL GROWTH OF DEGREE AT MOST k if
there is a finite constant C such that
|u(x)| <= C (1 + d_g(o,x))^k for every point x in M. (1.1)
The phrase "for every point x, simultaneously" is the whole subtlety.
A bound that holds only along some sequence of spheres going to infinity
does not satisfy (1.1). Every point must be controlled at once.
Let H_k(M,g) be the real vector space of all harmonic functions obeying
(1.1), and let
h_k(M,g) = dim_R H_k(M,g) (1.2)
be its dimension. This number counts how many independent harmonic
functions grow no faster than degree k. Our theorem is a strict
inequality between two such numbers.
1.4 The flat-space baseline.
Proposition 1.1 (harmonic polynomials in flat space). On R^n a harmonic
function of polynomial growth at most integer k is a harmonic polynomial
of degree at most k. Conversely every harmonic polynomial of degree at
most k obeys (1.1). Therefore
h_k(R^n, g_E) = sum_{l=0}^k N_l, N_l = C(m+l,m) - C(m+l-2,m), (1.3)
where m = n-1 and C(a,b) is the binomial coefficient "a choose b" (and
C(a,b) = 0 when b < 0 or b > a). Equivalently, after summing the
telescoping binomial identity C(a,b) - C(a-2,b) = C(a-2,b-1) +
C(a-1,b-1),
h_k(R^n, g_E) = C(n+k-1, k) + C(n+k-2, k-1). (1.4)
Proof. A homogeneous polynomial of degree l in n = m+1 variables has
dimension C(m+l, m). The Laplacian maps homogeneous polynomials of
degree l onto homogeneous polynomials of degree l-2 (it lowers degree
by exactly two). Its adjoint, under the natural inner product in which
distinct monomials are orthogonal, is multiplication by the squared
radius x_1^2 + ... + x_n^2, which is injective. Hence the Laplacian from
degree l to degree l-2 is surjective, and the HARMONIC polynomials of
degree l form a space of dimension
N_l = C(m+l, m) - C(m+l-2, m). (1.5)
Summing over l = 0,1,...,k gives (1.3). The identity
C(a,b) - C(a-2,b) = C(a-2,b-1) + C(a-1,b-1) is a one-line check from
the factorials, and applying it to (1.5) and summing gives (1.4).
It remains to see that "polynomial growth" forces "polynomial". Take a
harmonic u satisfying (1.1). For a smooth bump phi supported in the unit
ball with integral 1, set phi_R(x) = R^{-n} phi(x/R). The mean-value
property of harmonic functions says u is reproduced by convolution with
phi_R:
u(x) = int u(x + y) phi_R(y) dy.
Bounds (1.1) then bound |u(x)| in terms of R^k. Differentiating the
convolution in x moves all derivatives onto phi_R, giving
|partial^alpha u(x)| <= C_x R^k R^{-(k+1)}
for every multi-index alpha of order k+1. Letting R tend to infinity,
all derivatives of order k+1 vanish. A smooth function on R^n all of
whose (k+1)-st derivatives vanish is a polynomial of degree at most k.
Since u is harmonic, each homogeneous component is harmonic. This proves
the proposition. (The converse, that harmonic polynomials obey (1.1),
is immediate since a degree-l polynomial grows like distance^l.)
1.5 The question.
QUESTION. Suppose (M^n, g) is complete and has nonnegative Ricci
curvature (defined in Part 2; roughly, it does not bend inward on
average). Must we have
h_k(M,g) <= h_k(R^n, g_E) for every integer k >= 1? (1.6)
This is the integer-degree form of Yau's dimension comparison question.
It asks whether nonnegative Ricci curvature forces the count of
polynomial-growth harmonic functions to be no larger than the flat-space
count. The answer turns out to be NO. We build a counterexample for
some even n and some k, and an explicit one with n = 16, k = 50000.
1.6 Why the endpoint matters.
The analogous question at a REAL degree strictly between 1 and 2 was
already known to have a negative answer (Donnelly, and more recently
Cai-Lai, in dimensions at least five). But that does not settle the
integer endpoint: the flat-space space of quadratic harmonic polynomials
is strictly larger than its space at a degree between one and two. So a
counterexample at a fractional degree does not force a counterexample at
an integer. Our theorem closes precisely that gap under the Ricci
hypothesis alone, with no extra curvature assumption.
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PART 2. THE STRATEGY, IN PLAIN WORDS
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2.1 Ricci curvature.
For a unit vector v at a point, the Ricci curvature Ric(v,v) is the
average over all directions w perpendicular to v of the sectional
curvature of the plane spanned by v and w. Sectional curvature is the
way a small geodesic disk in that plane bends relative to flat space.
Ric >= 0 means: no direction, on average, bends inward more than flat
space does. It is the weakest of the classical lower curvature bounds
and the natural hypothesis for the counting question.
2.2 The shape we build.
We make a SPHERICALLY FIBRED space: a central point, then for each
radius r a copy of an odd-dimensional sphere S^m with a metric gamma(r)
on it, glued together with a radial coordinate r. Concretely we look at
metrics
g = dr^2 + r^2 gamma(log r), t = log r, (2.1)
where gamma(t) is a one-parameter family of metrics on S^m. When
gamma(t) is the round unit metric, (2.1) is exactly Euclidean space;
the origin is the point r = 0 and distance from the origin is r.
The family gamma(t) will be a VOLUME-NORMALIZED BERGER METRIC: the round
metric squashed along a circle direction, with the squash and the overall
scale allowed to vary slowly in t. The point of "volume-normalized" is
that the volume of each slice stays proportional to the round one, so
the space's total volume grows like r^n times a constant strictly between
0 and 1.
2.3 Why a moving link changes the count.
On a conical space g = dr^2 + r^2 h (a genuine cone with fixed link h),
a harmonic function that is an eigenfunction of the link Laplacian with
eigenvalue lambda has a growing mode proportional to r^d, where
d(d + m - 1) = lambda, d >= 0. (2.2)
So each angular direction of the link carries ONE fixed growth rate d.
If the link NEVER changes, each angular direction contributes at most
one growth rate, and the count is tightly constrained.
Our construction lets the link vary in t. A harmonic function can then
spend time at several different angular directions in succession, so
its growth rate is not a single one of the d's but a weighted average of
them over the cycle. If the space cycles through a collection of
directions whose AVERAGE growth rate is below an integer k, the function
still grows no faster than radius^k at every radius, yet it uses several
directions, and we can get strictly more independent functions than flat
space provides.
The hard part is making the succession EXACT. A tiny unwanted component
in a faster-growing direction will, after a long enough stretch of
radius, dominate everything else. So the switching must be arranged so
that it happens perfectly, with no leakage into unwanted directions.
That is what the control theory (Part 5) and the transmission estimates
(Part 8) accomplish.
2.4 The finite spectral surplus.
The angular directions live in the spaces V_l of round spherical
harmonics of degree l, for l = 2,3,...,L, with L finite. For each l we
choose an initial block of M_l directions (those with the smallest
growth rates at the nonround dwell metric). The spectral calculation
arranges integers 0 <= M_l <= N_l with
(1/M_l) sum_{i=1}^{M_l} d_{l,i} < k whenever M_l > 0, (2.3)
sum_{l=2}^L M_l > sum_{l=0}^k N_l = h_k(R^n, g_E). (2.4)
So the selected directions are more numerous than flat space allows
(2.4), but each block's AVERAGE growth rate is below k (2.3). If we can
make each harmonic function spend asymptotically equal logarithmic time
at all directions in its block, its overall growth rate will be the
block average, which is below k. Then all selected directions give valid
harmonic functions in H_k, and there are more of them than h_k in flat
space.
2.5 The course of the proof.
Part 3 derives the Berger link metrics and their Laplacian. Part 4
proves the spectral surplus (2.3)-(2.4) for large odd m, giving some
integer k automatically large; a separate explicit integer computation
at n = 16, k = 50000 is recorded in Part 10. Part 5 proves that short
angular deformations, using all complex structures J, allow us to steer
the angular directions at will: the trace-free squares of the Hopf
generators generate the full Lie algebra sl(V_l) in each degree. Part 6
builds the complete metric with nonnegative Ricci curvature, slowly
turning the link through squeeze/rest cycles. Part 7 sets up the
harmonic equation on this metric and identifies the space of
center-regular (smooth at the origin) harmonic functions. Part 8 proves
that, over a full cycle, the transmission of angular directions agrees
with a prescribed signed permutation pi_l to any desired accuracy, and
tunes every cycle exactly. Part 9 finishes: the exact cycling produces
sum M_l > h_k independent harmonic functions of growth at most k.
Part 9 also records the cone and volume conclusions.
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PART 3. THE BUILDING BLOCKS: BERGER LINKS
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3.1 The odd sphere and its Hopf direction.
Throughout the construction the angular link is an odd-dimensional
sphere S^m with m = 2s - 1, s >= 4. The reason we want m odd is that
the sphere S^m then carries a natural "Hopf" circle direction along
which we can squash it.
Concretely, regard S^m inside R^{m+1} = R^{2s}, and fix on R^{2s} an
ORTHOGONAL COMPLEX STRUCTURE: a real linear map J with
J^2 = -Id, J^T J = Id. (3.1)
Such a J is a 90-degree rotation on each complex line and is a
real orthogonal matrix, so it is an isometry of the round sphere. It
determines a vector field and a one-form on S^m:
xi_J(x) = J x, eta_J = g_0(xi_J, .), (3.2)
where g_0 is the round unit metric on S^m. Since xi_J is tangent to the
sphere and has constant length, it is a KILLING FIELD: its flow consists
of round isometries (rotations). The projection onto the line it spans is
P_J = xi_J \otimes eta_J. (3.3)
The DERIVATIVE along the Hopf field of a function u is
D_J u = du(xi_J). (3.4)
Because the flow of xi_J is a one-parameter group of isometries of the
round metric, the operator D_J is SKEW-ADJOINT for the round L^2 inner
product on each fixed sphere. This single fact underlies everything that
follows: the operator -D_J^2 = D_J^T D_J is symmetric and positive
semidefinite, and its eigenvalues are squares of the integer Hopf
charges (Part 4).
3.2 The volume-normalized Berger metric.
For two positive numbers a and q, define the metric
g_tilde_{q,J} = g_0 + (q-1) eta_J^2, (3.5)
which stretches or squashes the Hopf direction by the factor q while
leaving all perpendicular ("horizontal") directions alone. Then set
G(a,q,J) = a^2 q^{-1/m} g_tilde_{q,J}. (3.6)
There are two knobs. The parameter q controls the RATIO of the
Hopf-direction length to the horizontal length; the factor a controls
the overall scale. The extra power q^{-1/m} is chosen so that the volume
is a^m times the round volume, independent of q:
d vol_{G(a,q,J)} = a^m d vol_{g_0}. (3.7)
This VOLUME NORMALIZATION is essential in Part 6: it keeps the volume
growth of the whole space under control. All orthogonal complex
structures J are allowed; we never restrict to one component of the set
of such J.
3.3 Spherical harmonics.
For l >= 0, let V_l be the real vector space of restrictions to S^m of
homogeneous harmonic polynomials of degree l on R^{m+1}. These are the
SPHERICAL HARMONICS of degree l. Their dimensions are
N_l = dim_R V_l = C(m+l, m) - C(m+l-2, m), (3.8)
B_l = l(l+m-1). (3.9)
Here is why (3.8) holds, in case the same argument is wanted twice.
On homogeneous polynomials of every degree, give distinct monomials
x_1^{a_1}...x_n^{a_n} the orthogonal inner product with squared norm
a_1!...a_n!. The adjoint, in this inner product, of the Euclidean
polynomial Laplacian (from degree l to degree l-2) is multiplication by
the squared radius x_1^2 + ... + x_n^2, which is injective, hence the
Laplacian is surjective. So the harmonic polynomials of degree l have
dimension (number of degree-l monomials) - (number of degree-(l-2)
monomials) = C(m+l,m) - C(m+l-2,m), giving (3.8). Restriction from R^{m+1}
to S^m is injective on homogeneous polynomials, so V_l has that dimension.
Euclidean polar coordinates then show the round Laplacian on S^m acts on
V_l by the scalar B_l (3.9).
Every rotation of S^m preserves V_l. In particular, for every J the
derivative D_J preserves V_l and is skew-adjoint on it.
3.4 Lemma (Berger curvature and Laplacian). The metric G(a,q,J) of
(3.6) has volume form a^m dvol_{g_0}. Its Ricci endomorphism has two
eigenvalues: one on the (m-1)-dimensional horizontal space and one on
the vertical (Hopf) line, namely
horizontal: a^{-2} q^{1/m} ((m-1) - 2(q-1)),
vertical: a^{-2} q^{1/m} (m-1) q. (3.10)
Consequently the strict inequality Ric_{G(a,q,J)} > (m-1) G(a,q,J)
holds if and only if
a^2 < q^{1/m} min{ 1 - 2(q-1)/(m-1), q }. (3.11)
Call (3.11) the LINK-MARGIN inequality: it says the link is positively
curved by more than the round amount everywhere it will be used.
Every V_l is preserved by the positive Laplacian of G(a,q,J), and on
V_l that Laplacian is the operator
A_l(a,q,J) = a^{-2} q^{1/m} [ B_l Id + (q^{-1}-1) (-D_J^2 |_{V_l}) ]. (3.12)
Proof. The volume statement (3.7) is immediate: relative to g_0, the
metric g_tilde_q of (3.5) has eigenvalue 1 on the horizontal space and q
on the Hopf line, so its volume form is sqrt(q) dvol_{g_0}; multiplying
the metric by a^2 q^{-1/m} then multiplies the volume form by
a^m q^{-1/2}, giving a^m dvol_{g_0}.
For the curvature, write delta = q-1 (constant on the sphere). Let
nabla be the round connection, and set phi X = nabla_X xi. A direct
computation using the ambient derivative gives
phi X = JX + eta(X) x, phi xi = 0, phi^2 = -Id + xi \otimes eta,
(nabla_X eta)(Y) = g_0(phi X, Y), (nabla_X phi)Y = eta(Y) X - g_0(X,Y) xi. (3.13)
Let C = nabla_tilde - nabla be the difference between the connections
of g_tilde_q and g_0. The Koszul formula relates these, and because the
only non-round part of g_tilde_q is delta eta^2, one gets
C(X,Y) = delta( eta(X) phi Y + eta(Y) phi X ). (3.14)
In particular tr[X -> C(X,Z)] = 0, which simplifies the connection-change
formula for curvature to
(Ric_{g_tilde} - Ric_{g_0})(Y,Z)
= tr[X -> (nabla_X C)(Y,Z)] - tr[X -> C(Y, C(X,Z))]. (3.15)
These are traces of endomorphisms and can be computed in a round
orthonormal frame. Differentiating C in (3.14) and using (3.13), the
four terms contribute, in order,
-delta (g_0 - eta^2), delta (m-1) eta^2,
-delta (g_0 - eta^2), delta (m-1) eta^2, (3.16)
so the first trace is delta(-2 g_0 + 2m eta^2)(Y,Z). For the second
trace, putting (3.14) into itself and using phi xi = 0, phi^2 = -Id +
xi \otimes eta gives
C(Y, C(X,Z)) = delta^2 eta(Y)[ -eta(X) Z - eta(Z) X + 2 eta(X)eta(Z) xi ],
whose trace in X is -(m-1) delta^2 eta(Y) eta(Z). Since Ric_{g_0} =
(m-1) g_0, this yields the tensor identity
Ric_{g_tilde} = ((m-1) - 2(q-1))(g_0 - eta^2) + (m-1) q^2 eta^2. (3.17)
The squared length of xi in g_tilde_q is q, so the vertical eigenvalue
of the Ricci endomorphism is (m-1) q (the coefficient (m-1)q^2 divided
by the length q), while the horizontal eigenvalue is (m-1) - 2(q-1).
Constant scaling of the metric leaves the covariant Ricci tensor
unchanged and divides its endomorphism eigenvalues by the scale factor
a^2 q^{-1/m}, giving (3.10). Comparing both eigenvalues with (m-1) and
dividing through gives precisely (3.11).
Finally, the inverse metric is
G(a,q,J)^{-1} = a^{-2} q^{1/m} [ g_0^{-1} + (q^{-1}-1) xi \otimes xi ]. (3.18)
Because the volume form is a spatially constant multiple of the round
volume form, the divergence of a vector field is the same as the round
divergence. The Hopf field xi has zero round divergence and is a Killing
field, so div_{g_0}((D_J u) xi) = D_J^2 u. It follows that the positive
Laplacian on S^m satisfies
Delta_{G(a,q,J)}^{+} u
= a^{-2} q^{1/m} [ Delta_{g_0}^{+} u + (q^{-1}-1)(-D_J^2 u) ]. (3.19)
Restricting to V_l, where Delta_{g_0}^{+} = B_l Id, gives (3.12). This
proves the lemma.
3.5 What the two knobs do.
The parameter a only scales the whole sphere; it rescales the angular
eigenvalues by a^{-2} and does not distinguish directions. The parameter
q does something new: it TILTS the angular operator, weighting the
Hopf charge. As q moves away from 1, the eigenvalues of A_l on V_l
spread out, and different Hopf charges get different growth rates. This
is the mechanism that makes different angular directions grow at
different speeds, which in turn lets a harmonic function pick up
different rates by spending time in different directions. The next part
quantifies exactly how the eigenvalues are distributed.
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PART 4. THE SPECTRAL SURPLUS (FIRST HALF)
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Now we compute how the angular eigenvalues of the dwell metric are
distributed across directions, and convert them into growth rates.
4.1 Hopf charge and multiplicities.
Fix the standard complex structure J_0 on R^{2s} = R^{m+1}, with
coordinates z_1,...,z_s and their conjugates zbar_1,...,zbar_s, where
z_j = x_j + i y_j. A monomial that is homogeneous of DEGREE b in
z_1,...,z_s and of DEGREE l-b in zbar_1,...,zbar_s is said to have
bidegree (b, l-b). On such a monomial,
D_{J_0} acts by i(2b-l), (4.1)
because dx_j -> i dx_j and dy_j -> -i dy_j under J_0. So the square
-D_{J_0}^2 acts by the number (2b-l)^2. The integer 2b-l is the
HOPF CHARGE of the monomial. The possible Hopf charges in degree l are
-l, -l+2, ..., l-2, l.
The spaces V_l are preserved by D_{J_0}, so they split into charge
eigenspaces. Write g_l(b) for the real dimension of the charge-(2b-l)
eigenspace inside V_l. We need its exact value.
Lemma 4.1 (Hopf-weight multiplicity). For m = 2s-1 and 0 <= b <= l,
h_{l,b} := g_l(b)
= C(s+b-1, s-1) C(s+l-b-1, s-1)
- C(s+b-2, s-1) C(s+l-b-2, s-1) (4.2)
= (l+s-1)/(s-1) * C(b+s-2, s-2) * C(l-b+s-2, s-2). (4.3)
Terms with a missing bidegree in (4.2) count as zero. Contributions with
charges 2b-l and -(2b-l) are the same eigenvalue of -D_{J_0}^2 and are
ADDED; the middle charge when l is even is counted once.
Proof. Complexify V_l. A homogeneous polynomial of bidegree (b, l-b) in
the variables z and zbar maps under the operator
C := sum_j d^2/(dz_j dzbar_j) to bidegree (b-1, l-b-1). In the positive
monomial inner product, in which z^alpha zbar^beta has squared norm
alpha! beta!, the adjoint of C is multiplication by sum_j z_j zbar_j,
which is injective. Hence the space of HARMONIC polynomials of bidegree
(b, l-b) is the kernel of this surjective contraction, of dimension
dim (harmonic bidegree (b, l-b)) = (all bidegrees) - (the image)
= C(b+s-1, s-1) C(l-b+s-1, s-1)
- C(b+s-2, s-1) C(l-b+s-2, s-1), (4.4)
the first product counting bidegree-(b, l-b) polynomials in s complex
variables and the second the image in bidegree (b-1, l-b-1). This is the
first line. The second line is the difference of the two products
factored using Pascal's rule, and is an algebraic rearrangement.
Since D_{J_0} is real, its complex eigenspaces have complex dimension
equal to the real dimension of the underlying real eigenspace, so no
extra factor of 2 appears.
Summing (4.3) over b with the binomial convolution identity gives the
total dimension
N_l = (l+s-1)/(s-1) * C(l+2s-3, 2s-3)
= C(m+l,m) - C(m+l-2,m) ~ (2/(m-1)!) l^{m-1}. (4.5)
The multiplicities (4.3) are integers and are the exact weights used in the finite computer check of Part 10.
4.2 A beta distribution.
The numbers h_{l,b}/N_l (sum to 1 over b) behave like a beta-binomial
probability. This is the key to the counting argument. Concretely, if
X is a random variable on (0,1) with density proportional to
[x(1-x)]^{s-2} (a Beta(s-1, s-1) law), and, conditional on X, K_l is a
binomial random variable with parameters (l, X), then for each 0 <= b <= l
P(K_l = b) = h_{l,b} / N_l. (4.6)
This is the standard beta-binomial identity: integrating the binomial
probability C(l,b) x^b (1-x)^{l-b} against the beta density
x^{s-2}(1-x)^{s-2} up to normalizing constants gives exactly the ratio
(4.3) of beta functions. Since Var(K_l/l | X) = X(1-X)/l has expectation
at most 1/(4l), we have K_l/l -> X in probability under (4.6).
Now set
Y_m := (2X-1)^2. (4.7)
The change of variable y = (2x-1)^2 turns the density of X (proportional
to [x(1-x)]^{s-2}) into a density proportional to y^{-1/2}(1-y)^{s-2},
which is exactly the beta law with parameters (1/2, s-1) =
(1/2, (m-1)/2). Therefore
the numbers (2b-l)^2 / l^2, weighted by h_{l,b}/N_l, converge in
distribution as l -> infinity to Y_m ~ Beta(1/2, (m-1)/2). (4.8)
This is the distribution of squared normalised Hopf charge. It is
concentrated near 0 (small charge is common), with a density blow-up
at 0, and it controls how many directions have small growth rate.
4.3 From eigenvalues to growth rates.
Recall that at dwell parameters (a_*, q_*) the angular operator
A_l(a_*, q_*, J_0) acts on V_l by the scalars
lambda_{l,b} = alpha_*^{-2} [ B_l - (1 - 1/q_*) (2b-l)^2 ], (4.9)
where we used a_*^2 = q_*^{1/m} alpha_*^2 so that a_*^{-2} q_*^{1/m}
= alpha_*^{-2}, by (3.12). Order the eigenvalues increasingly as
lambda_{l,1} <= ... <= lambda_{l,N_l}, counted with real multiplicities,
and define the corresponding NONNEGATIVE growth rates d_{l,i} by
d_{l,i}(d_{l,i} + m - 1) = lambda_{l,i}, d_{l,i} >= 0. (4.10)
A short calculation: if d(d+m-1) = lambda and d >= 0, then
0 <= sqrt(lambda) - d <= (m-1)/2. (4.11)
Indeed d = sqrt(lambda + ((m-1)/2)^2) - (m-1)/2, so
sqrt(lambda) - d = sqrt(lambda) + (m-1)/2 - sqrt(lambda + ((m-1)/2)^2),
and squaring shows this lies in [0, (m-1)/2]. Thus d = sqrt(lambda) up
to a bounded additive error.
Since B_l / l^2 -> 1 and the empirical distribution of the squared
charges (2b-l)^2/l^2 converges to Y_m by (4.8), the increasing-order
growth rates behave, for 0 < v < 1, as
d_{l, ceil(v N_l)} / l
-> f_eps(v) := (1 - c eps)^{-1/2} sqrt( 1 - (1 - 1/q_*) y_m(v) ), (4.12)
where ceil is the ceiling (least integer at least its argument),
alpha_*^{-1} = (1 - c eps)^{-1/2}, and y_m(v) in (0,1) is the upper-tail
quantile of Y_m defined by P(Y_m > y_m(v)) = v. Convergence of the order
statistics to a quantile of a distribution with a continuous, strictly
increasing limiting law is standard; the rescaled d's are uniformly
bounded thanks to (4.11). Averaging the first ceil(u N_l) growth rates
for 0 < u < 1 gives
(1 / (l ceil(u N_l))) sum_{i=1}^{ceil(u N_l)} d_{l,i}
-> g_eps(u) := (1/u) integral_0^u f_eps(v) dv. (4.13)
The quantile formula (4.12) is the bridge: it tells us the growth rates
as a function of how far through the ordered list we go. Choosing an
initial block of a fraction u of the directions gives average growth
rate near l g_eps(u).
================================================================
PART 4. THE SPECTRAL SURPLUS (SECOND HALF)
================================================================
4.4 The weighted quantile inequality.
Recall Y_m ~ Beta(1/2, (m-1)/2) from (4.8). For 0 < v < 1 let y_m(v)
be its upper-tail quantile: the number in (0,1) with
P(Y_m > y_m(v)) = v. (4.14)
Define the weighted quantile integral
J_m := integral_0^1 y_m(v) log(1/v) dv. (4.15)
Lemma 4.2 (weighted quantile inequality). For all sufficiently large odd
m,
(m - 1) J_m > 2. (4.16)
This is the single numerical fact that drives the whole surplus.
Proof. Put r = m - 1. A standard representation of the beta law in terms
of independent normals is
r Y_m = Z^2 / (Z^2/r + W_r/r) in distribution, (4.17)
where Z is a standard normal variable and W_r is an independent sum of r
squared standard normals. Since E[W_r/r] = 1 and Var(W_r/r) = 2/r, we get
r Y_m -> Z^2 in distribution as r -> infinity. The law of Z^2 is
continuous and strictly increasing on (0, infinity), so the upper
quantiles converge: r y_m(v) -> y_inf(v), the upper quantile of Z^2, for
every 0 < v < 1. Fatou's lemma applied to (4.15) gives
liminf_{m->inf} (m-1) J_m
>= integral_0^1 y_inf(v) log(1/v) dv
= E[ R^2 log(1/Q(R)) ], (4.18)
where R = |Z| and Q(x) = P(R > x). The last equality is because the
strictly decreasing function Q makes Q(R) uniform on (0,1).
This expectation is finite. Since Q(R) is uniform on (0,1) and we want a
LOWER bound for log(1/Q(R)), we use an UPPER bound for Q. For x > 0,
integrating the half-normal density and using the classical Mills bound
gives
Q(x) = sqrt(2/pi) integral_x^inf e^{-t^2/2} dt
<= sqrt(2/pi) e^{-x^2/2} / x. (4.19)
Therefore 1/Q(R) >= R sqrt(pi/2) e^{R^2/2}, hence
R^2 log(1/Q(R))
>= R^4/2 + R^2 log(R sqrt(pi/2))
= R^4/2 + R^2 log R + (1/2) R^2 log(pi/2). (4.20)
Take expectations, writing A := E[R^2 log R]. Since E[R^2] = 1 (as R^2 =
Z^2 is standard-normal squared) and E[R^4] = E[Z^4] = 3,
E[R^2 log(1/Q(R))] >= 3/2 + A + (1/2) log(pi/2). (4.21)
Now we lower-bound A. Integration by parts against the half-normal
density gives E[R^4 log R] = 3A + 1. Put the probability measure
d nu = R^4 dP / 3 (its total mass is E[R^4]/3 = 1). Then
E_nu[log R] = E[R^4 log R] / 3 = A + 1/3. (4.22)
Jensen's inequality for the concave function log applied to 1/R gives
A + 1/3 = E_nu[log R] >= -log E_nu[1/R]. (4.23)
Since E_nu[1/R] = E[R^4 / R]/3 = E[R^3]/3 and E[R^3] = 2 sqrt(2/pi),
-log E_nu[1/R] = log(3 / E[R^3]) = log( 3 / (2 sqrt(2/pi)) ). (4.24)
Combining (4.21), (4.23), (4.24),
E[R^2 log(1/Q(R))] >= 7/6 + log(3 pi / 4). (4.25)
To finish, we check 7/6 + log(3 pi / 4) > 2 WITHOUT decimals. First,
pi/4 > 7/9: indeed arctan x > x - x^3/3 for x > 0 gives
arctan(1/2) > 11/24 and arctan(1/3) > 26/81, so
pi/4 = arctan(1/2) + arctan(1/3) > 11/24 + 26/81 = 505/648 > 7/9.
Hence 3 pi/4 > 7/3. Second, log(7/3) > 5/6: strict convexity of 1/t on
the two subintervals gives
log(7/3) = integral_1^{5/3} dt/t + integral_{5/3}^{7/3} dt/t
> (2/3)/(4/3) + (2/3)/2 = 1/2 + 1/3 = 5/6. (4.26)
Because 3 pi/4 > 7/3 and log is increasing, log(3 pi/4) > log(7/3) >
5/6, so 7/6 + log(3 pi/4) > 7/6 + 5/6 = 2. This proves (4.16).
4.5 The cutoff, the surplus lemma, and the count.
Before choosing the link we fix a smooth nondecreasing cutoff chi, equal
to 0 near (-infinity, 2] and equal to 1 near [4, infinity), and put
I_chi = integral_2^inf chi(t) (1+t)^{-5/4} dt > 0. (4.27)
The radial construction of Part 6 will ask only for the smallness
condition
( -log a_* ) / I_chi * sup_{t>2} [ chi(t) (1+t)^{-5/4} ] <= 1/4. (4.28)
This condition involves only the cutoff and the link scale a_*, before
any degree range or angular control has been chosen. It will make the
radial scale concave enough to keep Ricci curvature nonnegative in
Part 6, and since a_* can be taken arbitrarily close to 1 it can always
be met.
Lemma 4.3 (spectral surplus). There is an odd m = 2s - 1 >= 7 for which
the following is possible. The parameters a_* in (0,1) and q_* > 1 can
be taken arbitrarily close to 1, with the link-margin inequality (3.11)
holding at a = a_* for every q in [1, q_*]. For these parameters there
are integers k >= 2, L >= 2, and 0 <= M_l <= N_l for 2 <= l <= L such
that, writing d_{l,i} := d_{l,i}(a_*, q_*),
dbar_l := (1/M_l) sum_{i=1}^{M_l} d_{l,i} < k whenever M_l > 0, (4.29)
sum_{l=2}^{L} M_l > sum_{l=0}^{k} N_l = h_k(R^{m+1}, g_E). (4.30)
So the selected directions are more numerous than flat space allows
while each block's average growth rate stays below k.
Proof.
Step 1 (the link). Choose an odd m >= 7 satisfying Lemma 4.2, and fix
2/(m-1) < c < J_m. (4.31)
For a small positive parameter eps, set
q_* = 1 + eps, alpha_*^2 = 1 - c eps, a_*^2 = q_*^{1/m} alpha_*^2. (4.32)
For q >= 1, the right-hand side of the link-margin inequality (3.11) at
a = a_* is governed by
F(q) = q^{1/m} ( 1 - 2(q-1)/(m-1) ),
F'(q) = (m+1)/(m(m-1)) q^{1/m - 1} (1 - 2q) < 0. (4.33)
At q = q_*, F(q_*) = q_*^{1/m}(1 - 2 eps/(m-1)) while
a_*^2 = q_*^{1/m}(1 - c eps). Because c > 2/(m-1), we have
1 - c eps < 1 - 2 eps/(m-1), so a_*^2 < F(q_*). Since F is decreasing,
the strict inequality a_*^2 < F(q) holds for every q in [1, q_*], which
is exactly (3.11) restricted to a = a_*. Also a_*^2 < F(q_*) < F(1) = 1.
Step 2 (limiting means). By (4.12)-(4.13), the increasing-order growth
rates at (a_*, q_*) satisfy
d_{l, ceil(v N_l)} / l -> f_eps(v), (4.34)
(1/(l ceil(u N_l))) sum_{i=1}^{ceil(u N_l)} d_{l,i} -> g_eps(u), (4.35)
where f_eps(v) and g_eps(u) are as in (4.12)-(4.13). Define
I(eps) := integral_0^1 g_eps(u)^{-m} du. (4.36)
The functions stay bounded away from zero for small eps. Uniformly in
0 < v < 1 we have the expansion
f_eps(v) = 1 + (eps/2)( c - y_m(v) ) + O(eps^2), (4.37)
and the same uniform remainder holds after averaging over (0,u). Thus
g_eps(u) = 1 + (eps/2) (1/u) integral_0^u (c - y_m(v)) dv + O(eps^2),
so g_eps(u)^{-m} = 1 - (m eps/2) (1/u) integral_0^u (c - y_m(v)) dv +
O(eps^2). Integrate over u in (0,1) and use the identity
integral_0^1 (1/u) integral_0^u y_m(v) dv du
= integral_0^1 y_m(v) log(1/v) dv = J_m (4.38)
(the inner integral equals the region v <= u <= 1, then swap and
integrate 1/u from v to 1). This gives
I(eps) = 1 - (m eps/2)(c - J_m) + O(eps^2)
= 1 + (m eps/2)(J_m - c) + O(eps^2) > 1 (4.39)
for all sufficiently small eps > 0, because J_m > c. Fix such an eps,
as small as desired.
Step 3 (counting at integer thresholds). For each positive integer k and
l >= 2, let M_l(k) be the size of the largest initial block of the
ordered exponents whose mean is strictly less than k (and 0 if none
qualifies). The initial-block means are nondecreasing in the block
size, so "largest initial block with mean < k" is well defined. Only
finitely many degrees contribute, because (4.9)-(4.11) bound every
lambda_{l,i}, hence every d_{l,i}, below by a positive constant times
l^2, so min_i d_{l,i} grows at least linearly in l. The layer-cake
formula for an initial block is
M_l(k) = N_l integral_0^1
1{ (1/ceil(u N_l)) sum_{i=1}^{ceil(u N_l)} d_{l,i} < k } du. (4.40)
Fix u in (0,1) and delta > 0. By (4.35) the mean in the indicator is at
most l(g_eps(u) + delta/2) for all large l, hence is strictly below k
whenever l <= k/(g_eps(u) + delta) (including at the upper endpoint).
Using N_l ~ (2/(m-1)!) l^{m-1} from (4.5), the sum of the corresponding
N_l divided by sum_{l=0}^k N_l tends to (g_eps(u) + delta)^{-m} as the
integer k -> infinity. Let delta -> 0 and apply Fatou's lemma to (4.40):
liminf_{k->inf, k integer}
( sum_{l=2}^inf M_l(k) ) / ( sum_{l=0}^k N_l )
>= integral_0^1 g_eps(u)^{-m} du = I(eps) > 1. (4.41)
Discarding the finitely many lower degrees changes no limit.
Consequently every sufficiently large integer k has a strict surplus.
Take L to be the largest degree with M_l(k) > 0 and set M_l = M_l(k);
this gives (4.29)-(4.30).
A selected initial block may be empty, a singleton, or all of V_l, and
may cut through a repeated real eigenspace; when it does, choose the
indicated number of real orthonormal eigenvectors. The selected span
need not be invariant under any single D_J; only the full space V_l needs
to be invariant under the angular operators, which it is.
Finally, the count in (4.30) is exactly the Euclidean comparison
dimension in ambient dimension n = m+1:
sum_{l=0}^k N_l = C(m+k, k) + C(m+k-1, k-1) = h_k(R^{m+1}, g_E). (4.42)
For completeness, here is the direct Euclidean identification. Let u be
a Euclidean harmonic function of growth at most the integer k. A smooth
radial kernel psi_R(x) = R^{-(m+1)} psi(x/R), supported in the ball of
radius R and with integral 1, reproduces u by the mean-value property.
For each multi-index alpha of order k+1, moving derivatives onto the
kernel gives, at any fixed x and for R >= 1,
|partial^alpha u(x)| <= sup_{|y-x|<=R} |u(y)| * ||partial^alpha psi_R||_{L^1}
<= C_x R^k R^{-(k+1)} -> 0. (4.43)
Thus u is a polynomial of degree at most k, and its homogeneous
components are harmonic. This matches (4.42).
For the rest of the construction, "spectral data" means an odd
m = 2s - 1 with s >= 4, a fixed cutoff chi, and 0 < a_* < 1 < q_*
satisfying (3.11) throughout [1, q_*] and (4.28), together with finite
k, L >= 2 and 0 <= M_l <= N_l satisfying (4.29)-(4.30). These are
precisely the numerical hypotheses used in Parts 5 through 9. The
choices are made in the order m, c, eps, k, L; in particular, any
further requirement that a_* be close to 1 is imposed when eps is
chosen, before the integer cutoff k is fixed.
================================================================
PART 5. STEERING THE ANGLES: CONTROL BY COMPLEX STRUCTURES
================================================================
The plan of Parts 7-8 is this. During short "pulses" we briefly change
the squash q while keeping the angular metric close to round; the effect
on an angular direction is to apply the exponential of a trace-free
operator built from the Hopf generator D_J^2. Different pulses use
different complex structures J. If the collection of these operators is
rich enough, we can, with finitely many pulses, send any chosen angular
direction to any other. This part proves that richness (Proposition 5.1)
and packages it into a single finite "word" with an invertible
derivative (Corollary 5.2), plus the exact signed cycle Pi_l we shall
cycle.
5.1 Lie algebras, brackets, and generation.
For matrices A,B of the same size, the COMMUTATOR is
[A, B] = AB - BA. (5.1)
A LIE ALGEBRA of matrices is a real vector space of matrices closed
under [.,.]. The Lie algebra generated by a set S is the smallest such
space containing S; concretely it consists of all real linear
combinations of nested commutators of elements of S. The special linear
Lie algebra sl(V) of a real vector space V is the space of all
trace-free endomorphisms of V, i.e. of all linear maps V -> V whose
trace (sum of diagonal entries) is zero. It is a Lie algebra under
[.,.].
The importance here is that the FLOW of such an operator is a curve of
matrices exp(t A) in the group G = product over l of SL(V_l) (the
group of determinant-one linear maps of each V_l). If A, B generate a
big Lie algebra, then products of short flows exp(t_1 A) exp(t_2 B)...
can reach any element of the corresponding group, at least locally.
That is exactly what we need: a finite family of simple pulses whose
composition can be tuned (by choosing durations) to hit a prescribed
target in G.
5.2 The generators, and the statement.
Fix the spectral data of Part 4. Recall m = 2s - 1, s >= 4, and the
real spaces V_l for 2 <= l <= L, with dim N_l, each equipped with the
round L^2 inner product. For an endomorphism A of V_l write its
TRACE-FREE part
A_tf = A - (tr A / N_l) Id. (5.2)
For every orthogonal complex structure J on R^{2s}, the operator
D_J u = du(J x) (derivative along the Hopf field of J) preserves V_l and
is skew-adjoint, because it generates a one-parameter group of round
isometries. Therefore D_J^2 is symmetric negative semidefinite on V_l,
so (D_J^2)_tf is a real symmetric trace-free operator on V_l.
Proposition 5.1 (simultaneous control). Fix positive numbers kappa_l,
2 <= l <= L. As J ranges over all orthogonal complex structures on
R^{2s}, the tuples
X_J = ( kappa_l (D_J^2 |_{V_l})_tf )_{l=2}^{L} (5.3)
generate, under real linear combinations and commutators, the full Lie
algebra
direct sum_{l=2}^{L} sl(V_l). (5.4)
The positive kappa_l are just scalars; they do not change what can be
generated, only the shape of the generating family. Simultaneity means
that one single pulse, labeled by J, simultaneously applies a
trace-free generator in EVERY degree l at once. This is why a single
family of pulses can steer all degrees at the same time.
The proof takes several steps and is the longest in the document;
everything is elementary linear algebra once phrased on complexified
weight spaces.
5.3 Single degree: weight spaces.
We first prove (5.4) in a single degree l, with kappa_l = 1. Let
a be the real Lie algebra generated by (D_J^2)_tf on V_l, and put
W = V_l (tensor) C, and a_C = a (tensor) C the complexification. "Transpose"
in this section means the complex-bilinear extension of the real round
inner product; it is nondegenerate but not Hermitian.
Write orthonormal coordinates (x_1,y_1,...,x_s,y_s) and
z_i = x_i + i y_i. For theta = (theta_1,...,theta_s) in R^s, let
R_theta rotate coordinate plane i by theta_i, and let
T_theta u(x) = u(R_theta x). For nu = (nu_1,...,nu_s) in Z^s define
the WEIGHT SPACE of nu as
W_nu = { u in W : T_theta u = e^{ i (sum_i nu_i theta_i) } u
for every theta }. (5.5)
Then z_i has weight e_i and zbar_i has weight -e_i, and D_{J_0} acts
on W_nu by i sum_i nu_i. A monomial of holomorphic degree b and
antiholomorphic degree l-b has Hopf charge 2b - l, consistent with
Part 4. Every occurring weight obeys
sum_i |nu_i| <= l, l - sum_i |nu_i| in 2 Z. (5.6)
The EXTREME LAYER
Lambda_l = { nu in Z^s : sum_i |nu_i| = l } (5.7)
is especially simple: each such weight occurs, with a ONE-dimensional
weight space, spanned by the signed monomial z_i^{nu_i} for nu_i >= 0
and zbar_i^{-nu_i} for nu_i < 0 (a monomial of degree l), which is
harmonic since in each coordinate plane it involves at most one of
z_i, zbar_i, and is the only degree-l monomial of that weight. Interior
weights satisfying (5.6) may have dimension greater than one; we will
only need one-dimensional conclusions on the extreme layer.
5.4 Two symmetries.
(a) Torus invariance. Conjugating by a round rotation sends the family
(D_J^2)_tf of generators to itself, so a_C is invariant under the round
torus T_theta. A finite-dimensional subspace invariant under a torus
contains each of its Fourier components, because the Fourier projection
is an average of scalar multiples of torus conjugates. So a_C contains
every torus Fourier component (weight-shift piece) of each of its
elements.
(b) Sign averaging. For sigma = (sigma_1,...,sigma_s) in {+/-1}^s, let
J_sigma rotate coordinate plane i with sign sigma_i. On W_nu,
D_{J_sigma}^2 acts by -(sum_i sigma_i nu_i)^2. The elementary identity
2^{-s} sum_{sigma in {+/-1}^s} sigma_i sigma_j [ -(sum_a sigma_a nu_a)^2 ]
= -2 nu_i nu_j (5.8)
(for i != j, verified by expanding the square and noting that only the
i,j cross terms survive the sigma_i sigma_j weighting) shows that a_C
contains a diagonal operator A_ij that acts on the weight space W_nu by
A_ij : W_nu -> W_nu, v |--> (nu_i nu_j + c_ij) v, (5.9)
where c_ij is a constant independent of nu. The constant has no effect
on commutators, so for our purposes A_ij acts on weight nu by the
monomial function nu_i nu_j.
5.5 Isolating weight blocks.
A weight-shift endomorphism of shift delta in Z^s satisfies
T_theta A T_theta^{-1} = e^{i sum_i delta_i theta_i} A; equivalently it
maps each W_nu into W_{nu+delta} (with absent weight spaces read as 0).
On the block W_nu -> W_{nu+delta}, the commutator ad A_ij = [A_ij, .]
acts by the scalar
nu_i delta_j + nu_j delta_i + delta_i delta_j. (5.10)
Claim: if delta has at least three nonzero coordinates, these scalars
distinguish all source weights nu in a fixed shift component. Suppose
two sources give equal values, with difference v; then for all i != j
v_i delta_j + v_j delta_i = 0. (5.11)
On the support of delta, the ratios v_i/delta_i have pairwise sums zero,
and three such coordinates force every ratio to be zero; pairing a
coordinate outside the support with one inside then forces v_i = 0 there
too. This proves the claim.
There are only finitely many weights. Therefore, for a fixed shift
delta with at least three nonzero coordinates, polynomials in the
commuting operators ad A_ij can project onto an individual SOURCE
block within that shift: pick, for each competing source, one
coordinate of (5.10) that separates it from the desired source, and
multiply the corresponding linear interpolation factors. Polynomials
in ad A_ij preserve a_C. When both source and target lie on the extreme
layer Lambda_l, the isolated nonzero block is a RANK-ONE directed
matrix (one-dimension source, one-dimensional target).
5.6 From weight moves to a rank-one symmetric operator.
Regard an extreme weight as l SIGNED TOKENS (a signed point on the
lattice), with no opposite signs at the same coordinate. Call a
replacement a LEGAL MOVE if it replaces two equal-sign tokens at one
coordinate by one token at each of two OTHER coordinates (or the
reverse), both endpoints staying on Lambda_l. Such a shift has three
nonzero coordinates. We want a generator (D_J^2)_tf with a nonzero block
for that move.
Here is the explicit choice. For three distinct coordinate indices
i, j, k, let X_a, Y_a denote the coordinate vectors, and set
J X_i = X_j, J Y_i = X_k, J Y_j = Y_k, (5.12)
with the three reverse images determined by J^2 = -Id, and use the
standard complex structure on the remaining coordinate planes. This
defines an orthogonal complex structure. For signed variables
u_a^eps = x_a + i eps y_a with eps in {+/-1}, its Hopf field satisfies
D_J u_i^eps = -x_j - i eps x_k,
D_J u_j^beta = x_i - i beta y_k,
D_J u_k^gamma = y_i + i gamma y_j. (5.13)
For every choice of the three signs, the relevant replacement
coefficients are
u_i^eps -> u_j^beta : -1/2 u_i^eps -> u_k^gamma : -i eps/2
u_j^beta -> u_i^eps : 1/2 u_k^gamma -> u_i^eps : -i eps/2 (5.14)
There is no cancellation with other terms of D_J^2. Indeed, in the
independent complex-linear coordinates, the identity D_J^2 u = -u on
linear functions gives, on homogeneous degree-l polynomials,
D_J^2 p = -l p + sum_{u,v} (D_J u)(D_J v) d^2 p / du dv. (5.15)
If p = (u_i^eps)^a (u_j^beta)^b (u_k^gamma)^c M is an extreme monomial
(M using the other planes), the coefficient of the split monomial is
i eps a(a-1)/2 and that of the merge monomial is -i eps b c/2, nonzero
whenever the indicated move is legal: the exponent deficits of the
target force the two differentiated variables to be exactly those the
move specifies. Since rotations commute with the Euclidean Laplacian,
D_J^2 p is already harmonic, so no harmonic projection changes these
coefficients. Trace subtraction does not affect the shift component, and
Fourier projection together with the block isolation above converts
these into directed matrix units.
Next we join the two extreme weights -l e_1 and l e_1 by a simple path
of legal moves. For even l = 2h, use
-2h e_1 -> -(2h-2) e_1 + e_2 + e_3 -> ... -> h e_2 + h e_3
-> ... -> 2h e_1, (5.16)
splitting negative pairs at coordinate 1 in the first half and merging
the tokens at 2, 3 into positive pairs at 1 in the second half. This
includes l = 2, where the path is -2e_1, e_2 + e_3, 2e_1.
For odd l = 2h + 1, h >= 1, both starting signs admit the path
+/-(2h+1) e_1 -> ... -> +/-e_1 + h e_2 + h e_3
-> (h-1) e_2 + h e_3 + 2 e_4, (5.17)
where the last step merges the remaining token at 1 and one positive
token at 2 into two positive tokens at 4. Reverse the positive-start
path and append it to the negative-start path; the result is simple:
the two branches have opposite nonzero first coordinates until their
common endpoint, and no branch repeats a weight. All intermediate
weights remain extreme.
Choose monomial bases on these one-dimensional weight spaces and
normalize their directed matrices to matrix units. Nested brackets along
a simple path give its endpoint unit, because [E_{c,b}, E_{b,a}] =
E_{c,a} for distinct vertices. Hence a_C contains the unit from weight
-l e_1 to weight l e_1. Let v = z_1^l. By torus invariance of the
complex-bilinear round pairing, v pairs only with weight -l e_1, and
v^T vbar = integral_{S^m} |z_1|^{2l} dvol_{g_0} > 0. (5.18)
So the endpoint unit is a nonzero multiple of v v^T. Note also
v^T v = 0.
5.7 From rank-one to all trace-free matrices.
All round conjugates of v v^T belong to a_C. First, the round orbit of v
spans W. Up to a nonzero scalar, its members are the polynomials
(a . x)^l with a in C^{2s}, a != 0, a . a = 0 (real and imaginary parts
orthogonal and of equal length). On homogeneous degree-l polynomials
use the Fischer bilinear form
(p, q)_F = p(partial) q = sum_{|alpha|=l} alpha! p_alpha q_alpha. (5.19)
Its restriction to real harmonic polynomials is positive definite, so
its complex-bilinear extension to W is nondegenerate. If p in W
annihilates the orbit powers, then (p, (a . x)^l)_F = l! p(a) shows p
vanishes on the isotropic quadric, so p is divisible by Q(x) =
sum_a (x_a^2 + y_a^2). And a harmonic multiple of Q must vanish: in the
positive Fischer form, multiplication by Q is adjoint to the Euclidean
Laplacian, so if p = Q q and Laplacian p = 0 then
||p||_F^2 = <Q q, p>_F = <q, Laplacian p>_F = 0. (5.20)
Thus no nonzero element annihilates the orbit, proving it spans W.
For two orbit vectors u, w,
[u u^T, w w^T] = (u^T w) (u w^T - w u^T). (5.21)
For fixed u != 0 the factor u^T w is not identically zero on the orbit,
by nondegeneracy and the spanning property; as a real-analytic function
of the rotation it is nonzero on a dense set (the special orthogonal
group is connected). Dividing and passing to limits shows a_C contains
u w^T - w u^T for every pair of orbit vectors; these span the full
complex skew-symmetric algebra.
For every real skew-symmetric K, exp(ad K) preserves a_C. Hence a_C is
invariant under conjugation by the full real group SO(V_l). This group
sends v (up to scalar) to every nonzero isotropic vector w in W
(because their real and imaginary parts are orthogonal equal-length
pairs, and dim V_l >= 3). Consequently a_C contains w w^T for every
isotropic w. Those span all trace-free SYMMETRIC matrices: the rank-one
matrices from e_i + i e_j and e_i - i e_j give 2(E_ii - E_jj) as their
sum and 2i(E_ij + E_ji) as their difference in a real orthonormal basis.
Together with the skew-symmetric matrices just obtained, they span
sl(W). So a_C = sl(W), and since a is a real subspace of sl(V_l) with
dimension a_C = dimension a, a = sl(V_l). This proves (5.4) in one
degree.
5.8 Independence of the degrees.
We now combine the single-degree results into the simultaneous statement
(5.4). Let h be the Lie algebra generated by the tuples X_J of (5.3). The
one-factor projection onto sl(V_l) is full by whatever we just proved,
because scaling by kappa_l does not matter. Moreover
N_l = C(m+l,m) - C(m+l-2,m)
= C(m+l-1,m-1) + C(m+l-2,m-1) (5.22)
is strictly increasing in l. The real Lie algebras sl(V_l) are therefore
simple and pairwise nonisomorphic. In a real vector space of dimension at
least two, sl(V) is simple: in any nonzero ideal, a commutator with a
matrix unit first produces an element with an off-diagonal entry if
necessary; polynomials in the adjoint action of a generic trace-free
diagonal matrix then isolate an off-diagonal unit; commutators with the
other matrix units produce all off-diagonal units and all diagonal
differences. Hence an ideal of sl(V) is either 0 or all of sl(V). Distinct
dimensions of the V_l then give nonisomorphic sl(V_l).
Now we run a Lie-algebra analogue of the Goursat lemma, by induction on
the number of factors. Suppose the projection of h onto a preceding
product b (over smaller l) is full, and let c = sl(V_l) be the next simple
factor. The projected algebra h' sits inside b (+) c and surjects onto
both b and c. The kernel of its projection onto b is an ideal of c, and so
is either 0 or c. If it is c, the whole product b (+) c is present. If it
is 0, then h' is the graph of a surjective homomorphism b -> c. But the
image of each simple summand of b is an ideal of c, hence either 0 or c.
A nonzero image would make that summand isomorphic to c, contradicting
the fact that their dimensions differ. So every summand of b maps to 0,
impossible since the projection to c is surjective; that case cannot
occur. In either case, full product. The induction proves (5.4).
5.9 The signed cycle Pi_l.
We shall later ask the angular directions to cycle through the selected
block of M_l directions in each V_l. Fix, once and for all, the ordered
real orthonormal dwell eigenbasis e_{l,1},...,e_{l,N_l} of V_l (the
eigenbasis of the dwell angular operator, ordered by eigenvalue). If
M_l >= 1, define the signed cycle
Pi_l e_{l,i} =
e_{l,i+1} for 1 <= i < M_l,
(-1)^{M_l-1} e_{l,1} for i = M_l,
e_{l,i} for i > M_l, (5.23)
and set Pi_l = Id when M_l = 0. The sign of the M_l-cycle and the extra
column sign cancel, so det Pi_l = 1. Thus
Pi = (Pi_l)_{l=2}^{L} in G := product_{l=2}^{L} SL(V_l). (5.24)
This Pi is our target: over each full cycle we want the transmission of
angular directions to equal (up to a positive scalar and the diagonal
growth factor of Part 8) this Pi. Then the selected directions are
cycled exactly, and the growth-rate averaging of Part 9 works.
5.10 Corollary (a finite word with invertible differential).
Let d = sum_{l=2}^{L} (N_l^2 - 1). There exist finitely many orthogonal
complex structures J_1,...,J_R, an open neighborhood U of 0 in R^d, and
smooth real functions c_1,...,c_R on U such that
W(h) = exp(c_R(h) X_{J_R}) ... exp(c_1(h) X_{J_1}) (5.25)
satisfies W(0) = Pi and the derivative D W(0): R^d -> T_Pi G is an
isomorphism. The same assertion holds for any prescribed target in G.
Proof. Let H be the subgroup of G consisting of finite products of
exp(t X_J) with arbitrary real t. Define
E = span_R { Ad_h X_J : h in H }, (5.26)
where Ad_h X = h X h^{-1}. Conjugation by every generator exponential
preserves E; differentiating shows E is stable under bracketing with
each X_J. Since E contains the generators, it contains their generated
Lie algebra, which is the whole Lie algebra of G by Proposition 5.1.
Choose a basis Ad_{h_a} X_{K_a}, 1 <= a <= d, from this spanning set. The
map
F(t_1,...,t_d)
= (h_d exp(t_d X_{K_d}) h_d^{-1}) ... (h_1 exp(t_1 X_{K_1}) h_1^{-1})
(5.27)
has F(0) = Id and derivative
D F(0)(t) = sum_{a=1}^d t_a Ad_{h_a} X_{K_a}, (5.28)
which is an isomorphism by choice of basis. Each h_a and its inverse is
a fixed finite word of generator exponentials, so every value of F lies
in H.
The inverse function theorem gives an identity neighborhood contained
in H, so H is an OPEN subgroup. The group SL_N(R) is connected: polar
decomposition writes any positive-determinant matrix as a rotation times
a positive-definite determinant-one matrix, the latter joined to the
identity along S^t, 0 <= t <= 1, and SO(N) is connected. Hence G is
connected, and its open subgroup H is all of G.
Now choose a fixed finite word (in the form (5.25) with constant
coefficients) that represents the desired target Pi, and multiply it by
F. The product still has Pi as its value at zero, and its derivative
remains an isomorphism. Expanding the fixed conjugating words puts it
into the asserted form with fixed complex structures and smooth real
duration parameters. (The target Pi is fixed once the spectral data is
fixed, so this is legitimate before building any metric.)
5.11 Remarks on signs and orientation classes.
Two small points, both used later.
First, the real durations c_1,...,c_R may have either sign. They will be
realized, in Part 8, as integrals of bounded smooth bumps z, inside weak
metric pulses of the form q = 1 + z/T. Both signs are admissible once T
is large, because q > 0 and converges to 1 uniformly in the pulse.
Likewise, the class of J may be changed while q = 1, when the angular
metric is independent of J; this will let us concatenate pulses using
different complex structures without introducing curvature trouble.
Second, the quantifier over all orthogonal complex structures in
Proposition 5.1 includes BOTH orientation classes. This is crucial in the
sign averaging (5.8): with s = 4, restricting to one class would identify
some complementary sign characters and the averaging step would not be
valid. The explicit complex structure used for the token moves (5.12) can
itself be completed in either class by reversing it on an unused
coordinate plane. So no orientability obstruction arises.
================================================================
PART 6. A COMPLETE METRIC WITH NONNEGATIVE RICCI CURVATURE
================================================================
Now we build the space g = dr^2 + r^2 gamma(log r) promised in Part 2,
turning the link slowly enough that Ricci stays nonnegative. Fix the
spectral data of Part 4, so 0 < a_* < 1 < q_*, the link-margin
inequality (3.11) at a = a_* holds for every q in [1, q_*], and the
cutoff-smallness condition (4.28) holds. Fix also the reference complex
structure J_0 used for the dwell metric and for the last four pieces of
each period.
6.1 Pulse data.
Let K be a compact ball in a finite-dimensional real parameter space.
Choose finitely many disjoint open subintervals I_1,...,I_R of (0,1),
with disjoint closures, smooth functions beta_nu compactly supported in
I_nu, and orthogonal complex structures J_nu. For each nu let
c_nu(h) be smooth on a neighborhood of K. Set
z(tau, h) = sum_{nu=1}^{R} c_nu(h) beta_nu(tau). (6.1)
The amplitudes c_nu may have either sign. The functions beta_nu, the
complex structures J_nu, and the compact ball K are fixed once and for
all; they encode the "pulse family" from Corollary 5.10. In
applications the bumps are normalized to have integral 1, so their
amplitudes specify the durations c_1,...,c_R of the finite word.
6.2 The period schedule.
For j >= 1, define
T_j = j^6, L_j = j^7, t_1 = 10, t_{j+1} = t_j + L_j + 4 T_j. (6.2)
Choose a STARTING INDEX j_0, to be made large. Put q(t) = 1 for
t <= t_{j_0}. For each j >= j_0 choose a parameter h_j in K and divide
[t_j, t_{j+1}] into five successive intervals:
(1) a PULSE of duration T_j, on which
q(t) = 1 + T_j^{-1} z( (t - t_j)/T_j, h_j );
on the support of the nu-th bump, the complex structure is J_nu;
(2) an UPWARD RAMP of duration T_j, taking q from 1 to q_*;
(3) a DWELL of duration L_j, on which q = q_*;
(4) a DOWNWARD RAMP of duration T_j, taking q from q_* to 1;
(5) a ROUND REST of duration T_j, on which q = 1.
Throughout intervals (2)-(5) use J = J_0. Both ramps use a fixed smooth
nondecreasing profile rho:[0,1] -> [0,1], equal to 0 near 0 and 1 near 1;
their formulas in their own rescaled time are 1 + (q_* - 1) rho and
q_* - (q_* - 1) rho. The last four intervals form the DIAGONAL LEG of a
period. The notation J(t) is only a LABEL for the complex structure used
at time t; it need not be continuous. Labels are changed only in open
intervals where q = 1, and G(a,1,J) = a^2 g_0 is independent of J there.
The supported bumps provide such round gaps before, between, and after
their supports.
6.3 Proposition (the metric). For the fixed spectral data above, there
are a smooth function a: R -> (0,1] and an index j_geom such that, for
every j_0 >= j_geom and every sequence (h_j)_{j>=j_0} in K, the scheme
above defines a smooth complete metric on R^{m+1} with nonnegative Ricci
curvature,
g = dr^2 + r^2 gamma(log r), gamma(t) = G(a(t), q(t), J(t)). (6.3)
The function a is identically 1 for t <= 2, decreases to a_*, and for
some mu > 0 satisfies
b(t) := a'(t)/a(t) = -mu (1+t)^{-5/4} (t >= 4),
log( a(t)/a_* ) = 4 mu (1+t)^{-1/4} (t >= 4). (6.4)
The metric is Euclidean near the origin, its distance from the origin is
exactly r, and its volume form is
dvol_g = r^m a(log r)^m dr dvol_{g_0}. (6.5)
The function a and the starting index j_0 can be chosen independently
of the sequence of control parameters.
6.4 Proof, step one: the radial scale.
Use the fixed smooth nondecreasing cutoff chi from Part 4. For t > 2 put
b(t) = -mu chi(t) (1+t)^{-5/4}, a(t) = exp( integral_2^t b(u) du ), (6.6)
and set b = 0, a = 1 for t <= 2. The integral of chi(t)(1+t)^{-5/4} over
(2, infinity) is positive and finite, so there is a unique mu > 0 for
which a(t) tends to a_*; and mu -> 0 as a_* -> 1. Condition (4.28) gives
b >= -1/4
(because -log a_* = mu I_chi and the sup of chi(t)(1+t)^{-5/4} is at
least chi-valued; (4.28) is exactly b >= -1/4 at its minimum). (6.7)
Since chi' >= 0, for t > 2 we have b'(t) <= -5 b(t) / (4(1+t)), hence
b + b' + b^2 <= b ( 1 - 5/(4(1+t)) + b ) <= 0, (6.8)
because the last factor is positive for t > 2 and b >= -1/4. The tail
identities (6.4) follow by integrating b. This concavity (6.8) is the
one place the power 5/4 is used: it is the term that will keep the
radial Ricci component strictly positive.
6.5 Step two: the shape of a slice and the mixed Ricci tensor.
Write g_r = r^2 gamma(log r) for the metric on a slice at radius r, and
let a DOT denote differentiation in t = log r. Identify the tangent
bundles of the slices using the product coordinates. The SHAPE
ENDOMORPHISM (the second fundamental form of the slice in the whole
space) is
S = (1/2) g_r^{-1} partial_r g_r = r^{-1}(Id + B),
B = (1/2) gamma^{-1} dot gamma = b Id + H. (6.9)
Where a nonround part of a bump or a ramp occurs, J is fixed. Direct
differentiation of G = a^2 q^{-1/m}(g_0 + (q-1) eta_J^2) gives
H = (dot q / (2q)) (P_J - (1/m) Id),
tr H = 0,
tr(H^2) = (m-1)/(4m) (dot q / q)^2. (6.10)
On round gaps H = 0. Thus the same identities apply everywhere, without
requiring derivatives of the labels J(t).
The projection P_J is also the orthogonal projection for gamma onto the
Hopf direction. Indeed xi_J has constant squared length a^2 q^{1-1/m} on
a fixed slice, and its metric dual is the same constant times eta_J. The
flow of xi_J preserves both g_0 and eta_J, so xi_J is a Killing field for
gamma. A Killing field of constant length has zero divergence and
nabla_{xi_J} xi_J = 0 (the latter follows by pairing the Killing
equation with xi_J). Equivalently its unit normalization E satisfies
div_gamma (E (x) E^flat) = (div_gamma E) E^flat + (nabla^gamma_E E)^flat = 0.
Therefore div_gamma P_J = 0, and div_gamma H = 0, because the
coefficients in (6.10) are constant on a slice. The trace
tr S = r^{-1} m(1+b) is also spatially constant. The CODAZZI equation,
which compares the derivative of the shape in directions along the slice
with the derivative of the metric, therefore makes the MIXED Ricci tensor
(the Ricci entries one radial index, one angular index) identically zero.
6.6 Step three: the curvature identities.
For a metric dr^2 + g_r, the connection components involving the radial
variable are Gamma^r_{ij} = -(g_r)_{ik} S^k{}_j and Gamma^i_{rj} = S^i{}_j.
The NORMAL CURVATURE endomorphism (radial-radial direction) is
-partial_r S - S^2; its trace is Ric_g(partial_r, partial_r). Tracing the
tangential GAUSS equation contributes S^2 - (tr S) S to the intrinsic
slice Ricci endomorphism. Adding the normal contribution gives
Ric_g^# |_{TS^m} = Ric_{g_r}^# - partial_r S - (tr S) S, (6.11)
where an index of the tangential tensor is raised using g_r, and the
mixed formula is div_{g_r} S - d(tr S), already shown to vanish.
Substituting S = r^{-1}(Id + B) and tr B = m b yields
r^2 Ric_g(partial_r, partial_r) = -m(b + dot b + b^2) - tr(H^2),
r^2 Ric_g^# |_{TS^m} = Ric_gamma^# - dot B
- (m(1+b) - 1)(Id + B). (6.12)
In the tangential identity the H^2 terms cancel; only the concavity of a
and the positivity of the link survive.
6.7 Step four: uniform positivity on the periods.
All constants in the following estimates may depend on the fixed pulse
data and K, but not on j or on the choices h_j. On a pulse,
|q-1| = O(T_j^{-1}), |dot q| = O(T_j^{-2}), |ddot q| = O(T_j^{-3}). (6.13)
On either ramp, |dot q| = O(T_j^{-1}) and |ddot q| = O(T_j^{-2}); there
is no variation of q during the dwell or rest. Taking j_0 large makes
q > 0 throughout all pulses, including those with negative amplitudes;
the metrics on all slices then lie in a fixed uniformly equivalent
family. Formula (6.10) gives, throughout period j,
||H|| = O(T_j^{-1}), ||dot H|| = O(T_j^{-2}),
tr(H^2) = O(T_j^{-2}) = O(j^{-12}). (6.14)
The stronger pulse bound is tr(H^2) = O(T_j^{-4}). (To obtain the
derivative estimate, note that P_J is independent of t where q varies;
on every label-changing gap H vanishes identically.)
From (6.2),
t_j ~ j^8 / 8, sup_{t in [t_j, t_{j+1}]} | t/t_j - 1 | -> 0. (6.15)
On the scale tail, an exact calculation from (6.4) gives
-m(b + dot b + b^2)
= m mu (1+t)^{-5/4} ( 1 - 5/(4(1+t)) - mu (1+t)^{-5/4} ). (6.16)
This has a positive lower bound of order j^{-10}, uniformly on period j,
for all sufficiently large j. It dominates the O(j^{-12}) term tr(H^2),
so the radial component in (6.12) is positive once j_0 is large enough.
(Here we need the strict positivity from (6.8); this is where
nonnegative rather than merely bounded curvature is forced.)
For the tangential component, the strict link-margin inequality on the
compact interval [1, q_*] implies that for some fixed delta > 0,
Ric_{G(a_*,q,J_0)}^# >= (m-1+2 delta) Id (1 <= q <= q_*). (6.17)
(All J give the same intrinsic Ricci eigenvalues.) Since a(t) -> a_*,
this estimate persists with 2 delta replaced by delta along sufficiently
late ramps, dwells, and round rests. The pulse values satisfy q -> 1
uniformly; the strict inequality at (a_*,1) likewise supplies the same
conclusion on late pulses, after decreasing delta if necessary; this
argument allows q < 1 on a pulse. Thus all sufficiently late slices
satisfy Ric_gamma^# >= (m-1+delta) Id. On the other hand, B = b Id + H,
b, dot b -> 0, and the displayed bounds for H, dot H show uniformly that
dot B + (m(1+b)-1)(Id + B) = (m-1) Id + o(1). (6.18)
So the tangential component of (6.12) is positive as well. The
thresholds depend only on the fixed data and K. Since the mixed component
is zero, these estimates prove Ric_g >= 0 on every period, for every
admitted parameter sequence.
6.8 Step five: the initial region and the global space.
Before t_{j_0} the metric is round in the angular variable:
g = dr^2 + f(r)^2 g_0, f(r) = r a(log r). (6.19)
The scale construction gives
f'(r) = a(1+b) in (0,1],
f''(r) = (a/r)(b + dot b + b^2) <= 0. (6.20)
Its radial and tangential sectional curvatures are respectively
-f''/f and (1 - (f')^2)/f^2, as follows from the same normal curvature
and Gauss formulas. They are nonnegative, proving the required Ricci
condition in this region.
The pulse bumps and ramp profiles are constant or flat at all junctions,
and label changes occur only in round open intervals. Thus gamma, and
hence g, is smooth for r > 0. Only finitely many periods meet any
bounded interval of t, so the infinite sequence of periods introduces no
finite accumulation of junctions. For t <= 2, equivalently r <= e^2, both
a and q equal 1; the metric is exactly Euclidean there and extends
smoothly across the origin. The volume formula (6.5) follows from the
volume normalization (3.7).
A radial path from the origin to a point of radius r has length r.
Conversely, the length of any such path is at least the total variation
of its radial coordinate, and hence at least r. Thus d_g(0,x) = r(x).
The closed centered balls are the compact sets {r <= R} in the
underlying R^{m+1}, so the metric is complete. The underlying space is
connected, smooth, and without boundary. This completes the proof.
================================================================
PART 7. THE HARMONIC EQUATION AND THE IDEAL PULSE
================================================================
We turn now to the harmonic functions. On our metric the angular part
of a harmonic function stays inside a single space V_l, and its radial
part satisfies a matrix second-order equation. We set up that equation,
show it is governed by a bounded symmetric Riccati matrix, and record a
damping estimate used throughout Part 8.
7.1 Choosing the pulse word.
Let p_inf = m - 1. For 2 <= l <= L define
theta_l > 0, theta_l^2 + p_inf theta_l = a_*^{-2} B_l,
kappa_l = a_*^{-2} / (p_inf + 2 theta_l) > 0. (7.1)
These theta_l are the round growth rates at scale a_*: they are the
nonnegative solutions of d(d + m - 1) = a_*^{-2} B_l. The kappa_l is
exactly the Taylor coefficient that makes the linear response of the
angular operator match the radial response; it appears naturally in the
pulse analysis of Part 8.
Apply Corollary 5.10 to the target Pi of (5.24), using these kappa_l in
the generators (5.3). Restrict its parameter domain to a closed ball K
centered at zero and contained in U. Choose open intervals
I_1,...,I_R in (0,1), in chronological order with disjoint closures,
and smooth functions beta_nu compactly supported in I_nu with
integral_0^1 beta_nu = 1. Put
z(tau, h) = sum_{nu=1}^R c_nu(h) beta_nu(tau),
R_l(tau, h) = kappa_l sum_{nu=1}^R c_nu(h) beta_nu(tau)
( (B_l / m) Id + D_{J_nu}^2 |_{V_l} ). (7.2)
These functions, with all derivatives used below, are bounded on the
fixed compact K. Let H_l solve the IDEAL PULSE equation
partial_tau H_l = R_l(tau, h) H_l, H_l(0, h) = Id. (7.3)
For a matrix C on V_l with positive determinant, define the
determinant-normalized matrix
N_l(C) = (det C)^{-1/N_l} C. (7.4)
Each bump in R_l is a scalar function of time times one constant matrix,
so it contributes exp(c_nu(h) kappa_l (B_l Id / m + D_{J_nu}^2)) to
H_l(1,h), with later bumps multiplying on the left. For positive-
determinant matrices N_l(C_2 C_1) = N_l(C_2) N_l(C_1), and
N_l(e^C) = e^{C_tf} for every real matrix C. Removing the determinant
therefore removes the scalar part of each exponent. Consequently
Phi(h) := ( N_l(H_l(1,h)) )_{l=2}^{L} = W(h),
Phi(0) = Pi, D Phi(0) is an isomorphism. (7.5)
The determinants of the H_l are positive, since they are exponentials of
integrals of real traces. So Phi is a well-defined smooth map into the
group G, and by construction its derivative at 0 is an isomorphism (this
is the point of Corollary 5.10: after re-scaling by kappa_l, the ideal
word Phi agrees with W).
Apply Proposition 6.3 to this fixed pulse family. Its proof supplies a
threshold j_geom such that every j_0 >= j_geom and every parameter
sequence in K give the smooth complete metric g of (6.3) with
nonnegative Ricci curvature. Use exactly the radial scale a, cutoff, and
five-part schedule of Part 6, so d_g(0,x) = r, b = a'/a >= -1/4, and the
angular volume density is a(t)^m times the round volume. The four pieces
after the pulse form the diagonal leg shown in Part 6.
7.2 The angular part and the radial equation.
Fix l with 2 <= l <= L. The angular operator A_l(t) of (3.12) preserves
V_l. Hence a harmonic function u whose angular part lies in V_l can be
written
u(r, omega) = sum_{i=1}^{N_l} Y_i(log r) e_{l,i}(omega), (7.6)
and we identify its coefficient vector Y(t) = sum_i Y_i(t) e_{l,i} in
V_l. The function u is CENTER-REGULAR if it extends smoothly across the
origin r = 0. With t = log r, the harmonic equation becomes the matrix
second-order equation
Y'' + p(t) Y' - A_l(t) Y = 0,
p(t) = m - 1 + m b(t), A_l(t) = A_l(a(t), q(t), J(t)). (7.7)
Here is why. The radial volume density is r^m a(log r)^m; its
logarithmic derivative contributes m + m b to the radial equation, and
the change of variable t = log r subtracts 1, giving p = m - 1 + m b.
The density is constant in the angular variable, and the angular
operator A_l preserves V_l, so no additional angular terms or other
harmonic degrees occur.
7.3 Lemma (a damped matrix difference). Let E:[u,v] -> R^{N x N} be
continuously differentiable, and let F, L, R be continuous
matrix-valued functions on the same interval. Suppose
E' = F - L E - E R (7.8)
on [u,v], where L and R are SYMMETRIC and
lambda_min(L) + lambda_min(R) >= gamma > 0. Then, for u <= t <= v,
||E(t)|| <= e^{-gamma(t-u)} ||E(u)||
+ integral_u^t e^{-gamma(t-w)} ||F(w)|| dw, (7.9)
integral_u^v ||E(t)|| dt
<= gamma^{-1} ( ||E(u)|| + integral_u^v ||F(t)|| dt ). (7.10)
No commutativity of the matrices is assumed. Norms are Frobenius norms.
Proof. The Frobenius inner product gives
(1/2) d/dt ||E||^2
= <E,F> - <E, L E> - <E, E R>
<= ||E|| ||F|| - gamma ||E||^2, (7.11)
using symmetry of L,R and the lower bound on their smallest
eigenvalues. The corresponding upper differential inequality for ||E||
holds also at its zeros, by continuity (or by first replacing ||E|| by
(||E||^2 + eps^2)^{1/2} and letting eps -> 0). Multiplying by e^{gamma t}
and integrating gives (7.9). Integrating that estimate in t and
interchanging the nonnegative integrals gives (7.10).
7.4 Lemma (the center-regular value equation). There are a threshold
j_reg >= j_geom and constants 0 < c_P < C_P such that the following
holds for every j_0 >= j_reg, every sequence of parameters in K, and
every 2 <= l <= L. There is a unique smooth symmetric matrix P_l(t)
satisfying the RICCATI equation
P_l' = A_l - p P_l - P_l^2, P_l = l Id for t <= 2. (7.12)
It satisfies the two-sided barrier
c_P Id <= P_l(t) <= C_P Id for every t and 2 <= l <= L. (7.13)
The solutions of Y' = P_l Y are EXACTLY the center-regular solutions of
(7.7). Their value vector at any one time can be prescribed arbitrarily
and determines the solution uniquely. At each pulse start one also has
the reset bound
||P_l(t_j) - theta_l Id|| <= C j^{-2} (j >= j_0), (7.14)
including the first pulse. All constants and thresholds are uniform over
all preceding choices of parameters in K.
Proof. The matrices A_l(t) are symmetric. For sufficiently large allowed
j_0, the values of a and q lie in fixed compact positive intervals and
only finitely many V_l are involved, so (3.12) gives constants
0 < lambda_- Id <= A_l(t) <= lambda_+ Id. (7.15)
The radial bound b in [-1/4, 0] gives
0 < p_- := 3m/4 - 1 <= p(t) <= p_+ := m - 1. (7.16)
Choose c_P > 0 below all the initial values l so that
lambda_- - p_+ c_P - c_P^2 > 0, and choose C_P above all of them so that
lambda_+ - p_- C_P - C_P^2 < 0. We may decrease c_P and increase C_P so
that every theta_l also lies strictly between them.
The Riccati equation preserves symmetry. At a first contact with the
lower barrier, a unit vector v with P_l v = c_P v satisfies
v^T P_l' v >= lambda_- - p_+ c_P - c_P^2 > 0, (7.17)
and at a contact with the upper barrier the corresponding derivative is
at most lambda_+ - p_- C_P - C_P^2 < 0. These strict inward derivatives
prevent an eigenvalue from crossing either barrier. The bounds also keep
P_l in a compact set of matrices on every finite time interval, so local
existence for the ordinary differential equation continues for all
later times; uniqueness is local. In the Euclidean region, A_l = B_l Id
and p = m - 1, and P_l = l Id indeed satisfies (7.12) because
B_l - (m-1) l - l^2 = 0.
Let F_l(t) be the fundamental matrix of Y' = P_l Y with F_l(2) = Id. Its
determinant is
det F_l(t) = exp( integral_2^t tr P_l(w) dw ) > 0. (7.18)
Moreover Y'' = (P_l' + P_l^2) Y = A_l Y - p Y', so its columns solve
(7.7). On t <= 2 they are e^{l(t-2)} times constant vectors. Conversely,
the two scalar radial factors for (7.7) in this region are e^{l t} and
e^{-(l+m-1)t}; smoothness at the origin eliminates the second factor.
Thus every center-regular solution is one of the solutions Y' = P_l Y.
Invertibility of F_l(t) gives the assertion about arbitrary value data.
It remains to prove the reset bound (7.14). The schedule gives
t_j = j^8/8 + O(j^7), sup_{t in [t_j,t_{j+1}]} | t/t_j - 1 | -> 0. (7.19)
Times on period j and on the final rest immediately preceding it are
therefore comparable to j^8. From (6.4),
a(t) - a_* = O(j^{-2}), p(t) - p_inf = O(j^{-10}) (7.20)
uniformly on these intervals. On a round interval set E_0 = P_l -
theta_l Id. Subtracting (7.1) from (7.12) gives the exact difference
equation
E_0' = f_l^{rd}(t) Id - (p Id + P_l) E_0 - E_0 (theta_l Id),
f_l^{rd}(t) = a(t)^{-2} B_l - p(t) theta_l - theta_l^2 = O(j^{-2}), (7.21)
where the last bound uses (7.20). The symmetric coefficients p Id + P_l
and theta_l Id have a fixed positive sum of lower eigenvalue bounds.
Lemma 7.3, on a final rest of length T_{j-1}, gives
||P_l(t_j) - theta_l Id|| <= C e^{-gamma T_{j-1}} + C j^{-2}
= O(j^{-2}). (7.22)
The discrepancy at the start of the rest is bounded by (7.13), so it is
damped; it does not accumulate. For the first pulse, the interval up to
t_{j_0} is entirely round; apply the same estimate on
[t_{j_0}/2, t_{j_0}] once j_0 is large. That interval's length tends to
infinity and all its times are comparable to j_0^8, so the same bound
follows. This proves (7.14) with the claimed uniformity.
The matrix P_l is the logarithmic derivative of the value transfer of
center-regular solutions: if Y solves Y' = P_l Y with Y(t_0) = y, then
P_l(t_0) y = Y'(t_0). In Part 8 the bound (7.13) and the reset (7.14)
are the two inputs that make the pulse analysis uniform over all
histories.
================================================================
PART 8. EXACT TRANSMISSION OVER EACH PERIOD
================================================================
The set-up of the last two parts is now fixed: spectral data, the pulse
family, the word Phi with Phi(0) = Pi and invertible derivative, the
metric g with Ric >= 0, and the bounded symmetric Riccati matrix P_l with
reset bound (7.14). This part shows that a full period transmits angular
directions almost exactly as the ideal word, in a C^1 sense uniform in
all preceding controls, and then tunes each period's parameter h_j to
make the transmission EXACTLY a positive multiple of D_{l,j} Pi_l.
8.1 The response of one pulse.
Fix a period j, hold all parameters of earlier periods fixed, and let
the current parameter h vary in K. Write
T = T_j, tau = (t - t_j)/T, eta_j = j^{-2} + T^{-1}. (8.1)
Let B_{l,j}(h) be the VALUE TRANSMISSION for Y' = P_l Y from the start to
the end of the pulse, and write P_{l,e}(h) = P_l(t_j + T, h). Define the
SCALAR BASELINE solution
v_l' = a(t)^{-2} B_l - p(t) v_l - v_l^2, v_l(t_j) = theta_l,
varphi_{l,j} = integral_{t_j}^{t_j + T} v_l(t) dt. (8.2)
This scalar solution is independent of h and of the earlier parameters.
The number varphi_{l,j} is the log growth of the "round baseline" over
the pulse.
Lemma 8.1 (pulse limit with one parameter derivative). For the fixed
compact ball and pulse family,
e^{-varphi_{l,j}} B_{l,j}(h) = H_l(1, h) + O_{C^1}(eta_j), (8.3)
P_{l,e}(h) = theta_l Id + O_{C^1}(eta_j). (8.4)
Here O_{C^1}(eta_j) means that the norm and the norm of its first
derivative in the current parameter h, uniformly on K, are at most
C eta_j. The constants and the large-j threshold are independent of every
admitted preceding history.
Proof. The scalar barriers used in Lemma 7.4 also keep v_l between fixed
positive constants. Subtracting (7.1) from (8.2) gives
(v_l - theta_l)' =
( a(t)^{-2} B_l - p(t) theta_l - theta_l^2 )
- ( p + v_l + theta_l ) (v_l - theta_l). (8.5)
Its initial value is zero. The forcing is O(j^{-2}) by (7.20), and the
damping coefficient is uniformly positive. Thus
v_l - theta_l = O(j^{-2}) throughout the pulse. (8.6)
We retain v_l EXACTLY in the value equation: even the small pointwise
difference in (8.6) can have a large integral over T_j.
For the angular operator, expand (3.12) at q = 1. Its first derivative
in q is
a^{-2} ( (B_l / m) Id + D_J^2 |_{V_l} ). (8.7)
In particular the trace-free part has the positive coefficient
a^{-2} (D_J^2)_tf that appears in the control generators (5.3). Since
q = 1 + z/T on the pulse and a(t) - a_* = O(j^{-2}), Taylor's formula
gives
A_l(t, h) = a(t)^{-2} B_l Id + (p_inf + 2 theta_l)/T * R_l(tau, h)
+ E_{A,l}(t, h),
||E_{A,l}||_{C^1_h} <= C ( j^{-2}/T + 1/T^2 ). (8.8)
At most one bump is nonzero at any time; in the gaps the angular
operator is exactly scalar. Smooth boundedness of the profiles and their
first h-derivatives proves the uniform remainder bound, also across those
gaps. (The denominator p_inf + 2 theta_l is exactly what converts the
Taylor coefficient into the R_l used in (7.2).)
Set
Q_l(t, h) = v_l(t) Id + (1/T) R_l(tau, h). (8.9)
It is symmetric, and it has a fixed positive lower eigenvalue bound when
j is large. Its Riccati residual is
rho_l = A_l - p Q_l - Q_l^2 - Q_l'. (8.10)
The scalar terms in this expression cancel by (8.2). Using
partial_t( R_l / T ) = partial_tau R_l / T^2, the remaining identity is
rho_l = -( p - p_inf + 2(v_l - theta_l) )/T * R_l
- ( R_l^2 + partial_tau R_l ) / T^2 + E_{A,l}. (8.11)
Equations (8.6) and (8.8) therefore imply
sup_{t,h} ( ||rho_l|| + ||D_h rho_l|| ) <= C eta_j / T. (8.12)
Let E_l = P_l - Q_l. No matrices are commuted in the exact difference
equation
E_l' = rho_l - (p Id + P_l) E_l - E_l Q_l. (8.13)
The lower eigenvalue bounds for P_l, Q_l, and p allow Lemma 7.3 with a
fixed damping rate. The profiles vanish near tau = 0, so
E_l(t_j) = P_l(t_j) - theta_l Id = O(j^{-2}) (8.14)
by (7.14). From (7.9), (7.10), and (8.12), we obtain
sup_{t,h} ||E_l(t, h)|| <= C eta_j,
sup_h integral_{t_j}^{t_j+T} ||E_l(t, h)|| dt <= C eta_j. (8.15)
The initial discrepancy contributes its SIZE to this integral, because
it is damped; it does not contribute its size times T.
We next control the derivative in a current parameter. Derivatives are
taken with the earlier history fixed. Write E_{l,h} = D_h E_l and
Q_{l,h} = D_h Q_l; the calculation applies to every unit direction in
the finite parameter space. The coefficients of the Riccati equation
depend smoothly on h on the pulse, so the ordinary differential equation
has smooth parameter dependence there, and the uniform barriers keep its
solutions inside one fixed bounded set. Differentiating (8.13) and using
D_h P_l = E_{l,h} + Q_{l,h} gives the exact regrouping
E_{l,h}' = D_h rho_l - (p Id + P_l) E_{l,h} - E_{l,h} P_l
- Q_{l,h} E_l - E_l Q_{l,h}. (8.16)
Its initial value is zero: the past is fixed and R_l(0, h) = D_h R_l(0,h)
= 0. Also ||Q_{l,h}|| <= C/T. Apply Lemma 7.3 to (8.16). Its forcing has
pointwise norm at most C eta_j/T, by (8.12) and the first bound in
(8.15). Its integrated norm is at most
C eta_j + (C/T) integral_{t_j}^{t_j+T} ||E_l|| dt <= C eta_j. (8.17)
Consequently
sup_{t,h} ||E_{l,h}|| <= C eta_j / T,
sup_h integral_{t_j}^{t_j+T} ||E_{l,h}|| dt <= C eta_j. (8.18)
This supplies the derivative estimate directly, without an a priori
estimate for D_h P_l.
To pass from coefficient errors to value transmissions, let B_l(t,h) be
the pulse fundamental matrix up to time t, with value Id at t_j, and set
F_l(t,h) = e^{-integral_{t_j}^t v_l(w) dw} B_l(t,h),
G_l(t,h) = H_l( (t - t_j)/T, h ). (8.19)
Then
F_l' = (R_l/T + E_l) F_l, G_l' = (R_l/T) G_l,
F_l(t_j) = G_l(t_j) = Id. (8.20)
The integrated norms of R_l/T and D_h R_l/T are uniformly bounded; those
of E_l and E_{l,h} are O(eta_j). The elementary integral inequality
u(t) <= a + integral_{t_j}^t b(w) u(w) dw with b >= 0 implies
u(t) <= a exp( integral b ): the right-hand side has derivative at most b
times itself. Applying this to the fundamental-matrix equations and
their parameter derivatives bounds F_l, G_l, D_h F_l, D_h G_l uniformly.
For completeness, Z_l = F_l - G_l satisfies
Z_l' = (R_l/T + E_l) Z_l + E_l G_l, Z_l(t_j) = 0. (8.21)
The integral bound for E_l gives sup_t ||Z_l|| <= C eta_j. Differentiating
this equation introduces the additional terms
(D_h R_l/T + E_{l,h}) Z_l + E_{l,h} G_l + E_l D_h G_l, (8.22)
whose integrated norms are O(eta_j), using the preceding uniform bounds
and (8.18). The same integral inequality gives sup_t ||D_h Z_l|| <=
C eta_j. Evaluating at the pulse end proves (8.3). At that end
R_l(1,h) = D_h R_l(1,h) = 0. Combining P_l = Q_l + E_l, (8.6), (8.15),
and (8.18) proves (8.4). Every estimate used the earlier history only
through the uniform reset and barriers of Lemma 7.4, proving the stated
uniformity.
8.2 Removing the full diagonal leg.
The pulse estimate controls values and radial derivatives at the pulse
end. The radial derivative need not yet equal theta_l times the value,
so diagonal angular coefficients on the following leg do not by
themselves give a diagonal map on these data. We now factor the FULL
leg exactly. This separates its possibly large unequal growth from the
small error in the entrance derivative, and it is the step that makes
the control survive the long nonround dwell.
Let T_{l,j}(h) denote the value transmission of center-regular
solutions from t_j to t_{j+1}. On the diagonal leg (the four pieces after
the pulse) let U_{l,j} and V_{l,j} be the value propagation matrices
from the start to the end of the leg for initial Cauchy data
(Y, Y') = (Id, 0) and (0, Id), respectively. These matrices are
diagonal in the fixed dwell eigenbasis. Define
D_{l,j} = U_{l,j} + theta_l V_{l,j},
K_{l,j} = D_{l,j}^{-1} V_{l,j}. (8.23)
Thus D_{l,j} is the value propagation on the entire leg for the
reference data (Id, theta_l Id).
Lemma 8.2 (exact normalization of the full leg and dwell growth). The
matrices U_{l,j}, V_{l,j}, D_{l,j} have positive diagonal entries, and
0 <= K_{l,j} <= theta_l^{-1} Id. (8.24)
They depend only on the prescribed diagonal leg, not on its current or
earlier pulse parameters. The exact transmission identity is
D_{l,j}^{-1} T_{l,j}(h)
= [ Id + K_{l,j}( P_{l,e}(h) - theta_l Id ) ] B_{l,j}(h). (8.25)
Moreover, uniformly for the finite set of l,i,
log (D_{l,j})_{ii}
= L_j d_{l,i} + O(1 + L_j j^{-2} + T_j)
= L_j ( d_{l,i} + o(1) ). (8.26)
Proof. Every operator on the leg is a function of Id and D_{J_0}^2, and
hence is diagonal in the fixed dwell eigenbasis. A scalar coordinate
satisfies
y'' + p(t) y' - A_i(t) y = 0, A_i(t) > 0. (8.27)
Multiplying the equation for y' by the positive integrating factor
e^{integral p} gives
( e^{integral p} y' )' = e^{integral p} A_i y. (8.28)
Starting with (y, y') = (1, 0), positivity of y makes y' nonnegative
until any supposed first zero of y; this prevents that zero. Starting
with (0, 1), the solution is positive for small positive time and the
same argument keeps it positive. Thus the final entries of U_{l,j} and
V_{l,j} are positive. Formula (8.23) gives positivity of D_{l,j}, and
entry by entry
0 < (V_{l,j})_{ii} / ( (U_{l,j})_{ii} + theta_l (V_{l,j})_{ii} )
<= theta_l^{-1}. (8.29)
This proves (8.24).
For an incoming value vector y at the start of the period, the data at
the pulse end are
( B_{l,j} y, P_{l,e} B_{l,j} y ). (8.30)
Their value at the end of the leg is
T_{l,j} y = ( U_{l,j} + V_{l,j} P_{l,e} ) B_{l,j} y. (8.31)
Since D_{l,j} = U_{l,j} + theta_l V_{l,j}, left multiplication by
D_{l,j}^{-1} gives (8.25). The order K_{l,j}(P_{l,e} - theta_l Id) is part
of this identity; no commutation with P_{l,e} is used. The large matrix
D_{l,j} has disappeared from the error multiplier, leaving only the
uniform bound (8.24). This is the whole point of the diagonal leg: its
possibly huge growth is factored out exactly, and only a bounded
multiplier of the small entrance error survives.
To estimate D_{l,j}, take its i-th scalar solution with initial data
(1, theta_l). The positivity just proved allows its logarithmic
derivative w = y'/y throughout the leg. It satisfies
w' = A_{l,i}(t) - p(t) w - w^2. (8.32)
The scalar barrier argument in Lemma 7.4, with initial value theta_l,
keeps w between fixed positive constants. During the dwell,
A_{l,i}(t) = lambda_{l,i} + O(j^{-2}),
lambda_{l,i} = d_{l,i}( d_{l,i} + p_inf ), (8.33)
by (7.20) and the fixed value q = q_*. For e = w - d_{l,i}, subtraction
gives
e' = F_{l,i}(t) - ( p + w + d_{l,i} ) e,
F_{l,i} = A_{l,i} - lambda_{l,i} - (p - p_inf) d_{l,i} = O(j^{-2}). (8.34)
The value of e at the dwell entrance is bounded, and its damping
coefficient has a fixed positive lower bound. The scalar version of
(7.10) yields
integral_{dwell j} |w - d_{l,i}| dt <= C + C L_j j^{-2}. (8.35)
The two ramps and the final rest have total duration 3 T_j, and their
contribution to integral w is O(T_j). Since the solution starts at 1 and
ends at (D_{l,j})_{ii}, integration of w = (log y)' gives the first
equality in (8.26). The second follows from T_j / L_j = j^{-1} -> 0 and
L_j -> infinity. All constants are uniform over the fixed finite list of
coordinates.
8.3 Tuning every complete period exactly.
The preceding identity puts the actual full-period map in the same local
coordinate problem as the ideal word. Its remaining error tends to zero
with one parameter derivative, uniformly over the past. This allows the
target cycle to be attained successively in every period.
Proposition 8.3 (exact full-period transmission). For finite spectral
data as in Part 4, choose the word and compact pulse family of Part 7.
There are a starting index j_0 and parameters h_j in K, for every
j >= j_0, such that the center-regular value transmissions of the
resulting metric satisfy
T_{l,j} = s_{l,j} D_{l,j} Pi_l, s_{l,j} > 0,
|log s_{l,j}| <= C T_j (2 <= l <= L, j >= j_0). (8.36)
The positive diagonal matrices D_{l,j} propagate the reference Cauchy
data (Id, theta_l Id) over the entire leg following the pulse; they
satisfy the diagonal growth (8.26). The regular value equation has the
barriers (7.13). All estimates used to choose j_0 are uniform over
preceding admitted controls for this fixed finite family.
Proof. First fix an admitted past and vary the current parameter h. By
(8.25),
e^{-varphi_{l,j}} D_{l,j}^{-1} T_{l,j}(h)
= [ Id + K_{l,j}( P_{l,e}(h) - theta_l Id ) ]
e^{-varphi_{l,j}} B_{l,j}(h). (8.37)
The bound (8.24) and Lemma 8.1 imply
e^{-varphi_{l,j}} D_{l,j}^{-1} T_{l,j}(h)
= H_l(1, h) + O_{C^1}(eta_j). (8.38)
The estimate is uniform in the preceding history. Each T_{l,j} is a
value fundamental matrix of Y' = P_l Y, so it has positive determinant;
so does D_{l,j}. Determinant normalization is therefore defined for
their quotient. The matrices H_l(1,h), h in K, form a compact subset of
the positive-determinant matrices. On a neighborhood of this compact set,
N_l is smooth with bounded derivatives. Normalization is unchanged by
the positive scalar e^{-varphi_{l,j}}, so (8.38) gives
Psi_j(h) := ( N_l( D_{l,j}^{-1} T_{l,j}(h) ) )_{l=2}^{L}
= Phi(h) + O_{C^1}(eta_j) in G. (8.39)
The assertion is first an estimate for the ambient matrix entries and
their derivatives, and hence also in any fixed smooth chart containing
the compact image under consideration.
We spell out the uniform inversion. Use Euclidean norms on the parameter
and coordinate spaces, and the associated operator norms for their
derivatives. Choose a coordinate chart zeta near Pi in G with
zeta(Pi) = 0, and let f = zeta o Phi and A_0 = Df(0). By (7.5), A_0 is
invertible. Choose a closed ball B_r centered at 0, contained in the
interior of K, whose Phi-image lies compactly inside the chart and for
which
sup_{h in B_r} || Id - A_0^{-1} Df(h) || <= 1/4. (8.40)
For every sufficiently large j, uniformly in the admitted past,
f_j = zeta o Psi_j is defined on this ball and
sup_{h in B_r} || A_0^{-1}( Df_j(h) - Df(h) ) || <= 1/4,
|| A_0^{-1} f_j(0) || <= r/2. (8.41)
This follows from (8.39) and eta_j -> 0. The map
C_j(h) = h - A_0^{-1} f_j(h) (8.42)
has derivative of norm at most 1/2 on the ball. Also
|| C_j(h) || <= || C_j(0) || + (1/2) || h || <= r
for h in B_r. (8.43)
It is therefore a contraction of the closed ball into itself. Its
iterates converge to a fixed point h_j, for which f_j(h_j) = 0, that is,
Psi_j(h_j) = Pi.
All the preceding estimates were uniform over the past. Choose j_0 once,
large enough for them and for the geometric input, after the fixed word
and parameter ball have been chosen. The history before j_0 is round.
Choose h_{j_0} by the contraction above, and repeat after each finite
chosen history. This inductively defines an infinite admitted sequence,
hence a single smooth metric of Proposition 6.3. Undoing determinant
normalization at each fixed point gives
D_{l,j}^{-1} T_{l,j} = s_{l,j} Pi_l,
s_{l,j} = det( D_{l,j}^{-1} T_{l,j} )^{1/N_l} > 0. (8.44)
By (8.38), the determinant of e^{-varphi_{l,j}} D_{l,j}^{-1} T_{l,j}
stays bounded above and away from zero on the fixed compact family. Thus
log s_{l,j} = varphi_{l,j} + O(1) = O(T_j), (8.45)
because v_l is uniformly bounded. This proves (8.36). The equality is
for the complete period; no residual change of direction is passed to a
later period. This is what makes the cycling EXACT and prevents leakage
into faster-growing directions.
================================================================
PART 9. GROWTH, THE STRICT COUNT, AND THE CONES
================================================================
All parameters, the finite degree range, and the exact sequence of
controls are now fixed. We prove the pointwise growth bound, count the
independent harmonic functions, and then examine the cone limits at
infinity that explain why several comparison theorems do not apply.
9.1 Statement of the main theorem.
Theorem 9.1. There are an even integer n >= 8, an integer k >= 2, and a
complete smooth metric g on R^n with Ric_g >= 0 and
h_k(R^n, g) > h_k(R^n, g_E), (9.1)
the Euclidean count (1.4). The metric is Euclidean near the origin, and
its asymptotic volume ratio satisfies
0 < lim_{R->inf} vol_g B_g(0,R) / (omega_n R^n) < 1, (9.2)
where omega_n is the volume of the Euclidean unit ball. Concretely one
may take n = 16 and k = 50000.
9.2 The selected harmonic functions.
For each l with M_l > 0 and each 1 <= i <= M_l, choose the center-regular
solution with value vector Y(t_{j_0}) = e_{l,i}. Lemma 7.4 permits this
choice because it lets us prescribe the value vector at any one time.
Near the origin it is r^l times an element of V_l, hence a homogeneous
harmonic polynomial in Euclidean coordinates; it is smooth and harmonic
there, and, by (7.7), on the rest of the space.
Let sigma_l be the cyclic permutation of {1,...,M_l} underlying Pi_l,
and define i_{j_0} = i, i_{j+1} = sigma_l(i_j). Applying the exact
transmission (8.36) sends e_{l,i_j} to a signed multiple of
e_{l,i_{j+1}}, with absolute multiplier
s_{l,j} (D_{l,j})_{ i_{j+1}, i_{j+1} }. (9.3)
Hence, with
varepsilon_{l,i,j} = L_j^{-1} log(D_{l,j})_{ii} - d_{l,i},
max_{l,i} | varepsilon_{l,i,j} | -> 0 (9.4)
by the diagonal growth (8.26), one has
log ||Y(t_J)|| = sum_{j=j_0}^{J-1}
[ log s_{l,j} + j^7 ( d_{l, i_{j+1}} + varepsilon_{l, i_{j+1}, j} ) ]. (9.5)
Signs in Pi_l do not affect this norm. So the growth of the selected
harmonic function is governed by the cycle average of the rates d_{l,i}
and by the small per-period factors s_{l,j}.
9.3 The weighted average over cycles.
We compute the leading behavior of (9.5). Group the sum of
j^7 d_{l,i_{j+1}} into complete cycles of length M_l. Within a cycle
starting at j, (j+r)^7 - j^7 = O(j^6) for 0 <= r < M_l, with a constant
depending on the fixed length. Since the sum of d_{l, i_{j+r+1}} - dbar_l
in a complete cycle is zero, that cycle contributes only O(j^6) to the
difference from the mean-weighted sum. Summing these errors, and bounding
the at most M_l - 1 final terms by O(J^7), gives
sum_{j=j_0}^{J-1} j^7 d_{l,i_{j+1}}
= dbar_l sum_{j=j_0}^{J-1} j^7 + O(J^7). (9.6)
Also, by (8.36), sum_{j<J} |log s_{l,j}| = O( sum_{j<J} j^6 ) = O(J^7).
The varepsilon-terms sum to o(J^8): for any positive number, choose a
fixed late index after which their absolute values are below that
number, and use sum_{j<J} j^7 = O(J^8); the finitely many earlier terms
vanish after division by J^8. Finally,
sum_{j=j_0}^{J-1} j^7 = J^8/8 + O(J^7),
t_J = J^8/8 + O(J^7) (9.7)
by (6.2). Therefore
lim_{J->inf} log ||Y(t_J)|| / t_J = dbar_l < k. (9.8)
The limit is strictly below k because of the spectral selection (4.29).
9.4 Extending to every radius.
The estimate at the period endpoints t_J extends to every radius. From
Y' = P_l Y and the Riccati barrier (7.13),
||Y(t)|| <= exp( C_P (t - t_j) ) ||Y(t_j)|| (t_j <= t <= t_{j+1}). (9.9)
The ratio (t_{j+1} - t_j)/t_j tends to zero by (7.19). Thus (9.8) gives
limsup_{t->inf} log ||Y(t)|| / t <= dbar_l. (9.10)
Choose alpha_l with dbar_l < alpha_l < k. For all sufficiently large
r = e^t, ||Y(log r)|| <= r^{alpha_l}. The fixed finite-dimensional space
V_l has
| sum_i Y_i e_{l,i}(omega) | <= C_l ||Y||
for every omega in S^m, (9.11)
for example by Cauchy-Schwarz and boundedness of the finitely many
smooth basis functions. Since d_g(0,x) = r(x), this gives
|u(x)| <= C_u (1 + d_g(0,x))^k (9.12)
outside a compact set. Smoothness bounds u on that compact set and
supplies the same inequality everywhere. So each selected u lies in
H_k(R^{m+1}, g).
9.5 The strict count and the volume ratio.
These functions are linearly independent: their restrictions to the
sphere t = t_{j_0} are the selected vectors of mutually orthogonal round
spaces V_l. It follows that
dim_R H_k(R^{m+1}, g) >= sum_{l=2}^{L} M_l > sum_{l=0}^{k} N_l. (9.13)
The Euclidean identification (4.42) shows the right-hand side is
h_k(R^{m+1}, g_E), so this is the required strict comparison (9.1).
Finally, the distance and volume statements of Proposition 6.3 give
vol_g B_g(0,R)
= vol_{g_0}(S^m) integral_0^R r^m a(log r)^m dr. (9.14)
Because a(t) -> a_*, changing variables r = R z and using dominated
convergence shows that this quantity divided by omega_{m+1} R^{m+1}
tends to a_*^m; here vol_{g_0}(S^m) = (m+1) omega_{m+1}. Since 0 < a_* <
1, the asymptotic volume ratio (9.2) lies strictly between 0 and 1. This
completes the proof of Theorem 9.1.
9.6 Cone geometry, conformal flatness, and cone degrees.
Proposition 9.2 (cone consequences). Let g be the metric constructed in
Theorem 9.1, with link parameters a_*, q_* and comparison integer k.
(a) Every metric cone dr^2 + r^2 G(a_*, q, J_0), 1 <= q <= q_*, occurs
as a pointed tangent cone of (R^{m+1}, g) at infinity. In particular the
tangent cone is NOT unique.
(b) The metric g is NOT locally conformally flat.
(c) Let D be the union of all nonnegative homogeneous harmonic degrees
over all tangent cones of (R^{m+1}, g). Then D contains a half-line and
a neighborhood of k.
Proof. (a) Center a fixed logarithmic window [log R - A, log R + A],
with A fixed, inside either a round rest (q = 1) or a nonround dwell
(q = q_*), and let the center log R tend to infinity. Because the
schedule is self-similar with slowly growing periods, this produces
smooth annular limits whose links are G(a_*, 1, J_0) and
G(a_*, q_*, J_0). These two links are nonisometric: the first is
Einstein (all Ricci eigenvalues equal), whereas the second has distinct
horizontal and vertical Ricci eigenvalues by (3.10). Every intermediate
link G(a_*, q, J_0), 1 < q < q_*, also occurs, because it appears on the
ramps and dwells, whose durations L_j diverge while their time
derivatives tend to zero. These annular limits extend to pointed cone
limits including the vertex: the inner radial ball of radius delta has
diameter at most 2 delta in every rescaling and in the limit; comparing
paths across the complementary annulus, then letting delta -> 0, gives
convergence of distances on each bounded ball. All these links have the
same volume a_*^m vol_{g_0}(S^m); in particular the family of tangent
cones is not unique.
(b) The nonround cone metric dr^2 + r^2 h_*, with h_* = G(a_*, q_*, J_0),
is conformal, under t = log r, to the product metric dt^2 + h_*. For a
product metric, the WEYL CONFORMAL CURVATURE tensor has mixed
components
W_{0a0b} = -1/(m-1) ( Ric_{h_*} - (Scal_{h_*}/m) h_* )_{ab}. (9.15)
Indeed, the product has zero mixed curvature and radial Ricci tensor;
substituting in the trace decomposition of curvature gives (9.15). This
component is nonzero because the link has distinct horizontal and
vertical Ricci eigenvalues by (3.10). Vanishing of the Weyl tensor is a
conformally invariant condition, and it is preserved by smooth annular
limits. If the constructed metric were locally conformally flat, then
its nonround cone limit would have zero Weyl tensor, contradicting
(9.15). Hence g is not locally conformally flat.
(c) For maximal Hopf charge in degree l, its growth root alpha_l(q) is
the nonnegative solution of
alpha_l(q)( alpha_l(q) + m - 1 )
= a_*^{-2} q^{1/m} [ l(l+m-1) + (q^{-1} - 1) l^2 ]. (9.16)
This root varies continuously with q. At q = 1 its asymptotic slope in
l is a_*^{-1}; at q = q_* the slope is a_*^{-1} q_*^{-(m-1)/(2m)}, which
is strictly smaller. More precisely, the endpoint roots are
alpha_l(1) = a_*^{-1} l + O(1),
alpha_l(q_*) = a_*^{-1} q_*^{-(m-1)/(2m)} l + O(1). (9.17)
The difference of the slopes is positive, so for every sufficiently
large l, alpha_{l+1}(q_*) < alpha_l(1) and alpha_l(q_*) < alpha_l(1).
The continuous image of the q-interval contains the entire interval
between these endpoints. These intervals overlap and have unbounded
upper endpoints, so their union contains a half-line. In particular,
equal cone-link volumes do not provide a discrete union spectrum.
The chosen integer k itself lies in D. Indeed, the spectral selection
(4.30) forces some selected degree l > k. Its smallest dwell root
d_{l,1} is below k (by (4.29)), while its round root theta_l is greater
than l because a_* < 1. Continuity of alpha_l(q) in q therefore supplies
an intermediate cone root equal to k; the strict endpoint inequalities
in fact put a whole neighborhood of k in D. Xu's nonunique-cone
three-circles theorem requires the comparison exponent to lie OUTSIDE
D, so it cannot be applied at k or at any sufficiently nearby exponent.
The unique-cone spectral bounds of Huang also require a hypothesis that
fails here: a global logarithmic growth average, as in (9.8), need not
equal the local homogeneous degree seen after normalization on each
separate cone subsequence. This proves the proposition.
This is the sense in which the construction is a genuine boundary case
for the cone-spectral theory: the moving link produces many tangent
cones with a range of spectra, and the integer cutoff k lands in that
range rather than outside it.
================================================================
PART 10. AN EXPLICIT FINITE SELECTION: n = 16, k = 50000
================================================================
The analytic argument of Part 4 supplies SOME spectral data, with k
automatically large. A supplementary integer calculation pins down
concrete numbers. It checks only the finite spectral inequalities
(4.29)-(4.30); it does not construct the finite control word or the
metric numerically. The analytic existence argument is independent of it.
10.1 The parameters.
The parameters of the calculation are
m = 15, s = 8, k = 50000, q_* = 101/100,
c = 147/1000, alpha_*^2 = 99853/100000,
a_*^2 = q_*^{1/15} alpha_*^2. (10.1)
The endpoint curvature inequality (3.11) is strict because
1 - 2(q_* - 1)/(m - 1) - alpha_*^2 = 29/700000 > 0. (10.2)
The integer inequality 101 * 99853^15 < 100 * 100000^15 shows that
q_* (alpha_*^2)^15 < 1, and hence a_* < 1. The decrease of the curvature
threshold in the proof of Lemma 4.3 gives the margin for every
q in [1, q_*]. The radial scale meets its required smallness (4.28):
-log a_* <= -(1/2) log alpha_*^2
< (1 - alpha_*^2)/(2 alpha_*^2) < 1/1000. (10.3)
The cutoff integral in Proposition 6.3 is at least
integral_4^inf (1+t)^{-5/4} dt = 4 * 5^{-1/4} > 2, so (4.28) gives
mu < 1/2000, in particular b >= -1/4. These are parameter checks; the
control and transmission arguments remain the analytic arguments of
Parts 5 through 9.
10.2 The integer bounds.
Write h = m - 1 = 14, Q = 10^8, and an angular eigenvalue as
lambda = num/den > 0, with integer numerator and denominator. If
isqrt(a) = floor(sqrt a) for a nonnegative integer a, define
S = isqrt( floor( Q^2 ( h^2 den + 4 num ) / den ) ) + 1. (10.4)
Then S^2 den > Q^2 ( h^2 den + 4 num ), so the positive root of
d(d + h) = lambda satisfies the strict rational bound
d < ( S - h Q ) / ( 2 Q ). (10.5)
For the charge indexed by b, the exact eigenvalue data are
num = 100000 ( 101 l(l+14) - (2b - l)^2 ),
den = 101 * 99853. (10.6)
The multiplicities are the integers h_{l,b} of (4.2). The two equal
contributions indexed by b and l-b are combined, except for the middle
term when they coincide. Increasing b from 0 to floor(l/2) puts the
eigenvalues, and therefore the upper bounds for the exponents, in
increasing order.
The program verification/counting-check.py uses these integer
multiplicities and upper bounds. It accepts an initial group only when
its upper-bounded exponent sum is strictly less than k times its size. It
may take part of a repeated eigenvalue, which corresponds to choosing
that many vectors in its real eigenspace. Low degrees are selected in
full whenever the exact inequality l(l+14) < alpha_*^2 k(k+14)
certifies all their roots to be below k. In odd degree the zero Hopf
charge is absent, so the full-degree test is a sufficient criterion
rather than an exact characterization of the largest root. The individual
search ends when the smallest root is at least k. Thus every counted
group has a true mean strictly below k.
10.3 The resulting counts.
Using integer arithmetic, the counts are
selected dimension
= 46785605044474340387088811257657817010714687366486217934111,
h_50000(R^16)
= 46779709349146362239266126538609505511855492134986686636251,
strict surplus
= 5895695327978147822684719048311498859195231499531297860. (10.7)
The selected dimension omits degrees 0 and 1, as required in (4.30).
Only the displayed strict integer comparison is used by the certificate;
a decimal ratio is unnecessary.
These parameters and selections satisfy all the finite-data hypotheses
at the end of Part 4. Applying Proposition 6.3 (the metric with Ric >= 0)
and Proposition 8.3 (exact transmission), followed by the growth
argument of Part 9, therefore gives the metric of Theorem 9.1 with
n = 16 = m+1 and k = 50000. This completes the proof of the main
theorem at the stated explicit numbers.
================================================================
GLOSSARY
================================================================
BERGER METRIC. A round sphere metric squashed along a Hopf circle
direction; here G(a,q,J) = a^2 q^{-1/m}(g_0 + (q-1) eta_J^2). The
parameter q controls the squash, a the scale, J the circle direction. We
use the volume-normalized version, so the volume does not depend on q.
BETA DISTRIBUTION. The law on (0,1) whose density is proportional to
[x(1-x)]^{s-2} for Beta(s-1,s-1), and to y^{-1/2}(1-y)^{s-2} for
Beta(1/2,(m-1)/2). Here Y_m = (2X-1)^2 has the second form.
CENTER-REGULAR. A harmonic function on the whole space that extends
smoothly across the origin. These are exactly the solutions of Y' = P_l Y
for the Riccati matrix P_l.
COMPLETE METRIC. A metric in which every path of finite length that
ought to converge does; equivalently, distance balls are compact.
Our metric is complete because every distance ball is a compact ball in
R^{m+1}.
CURVATURE.
Sectional curvature: bending of a small geodesic disk in a 2-plane,
relative to flat space.
Ricci curvature Ric(v,v): average of sectional curvatures over planes
containing v and each perpendicular direction.
Nonnegative Ricci curvature: no direction bends inward more than flat
space, on average.
DIRECT SUM. The space of tuples (A_2,...,A_L) acting independently in
each V_l. sl(V_l) is the trace-free endomorphisms of V_l; the direct sum
of the sl(V_l) is the Lie algebra whose flows steer all degrees at once.
FISCHER FORM. The bilinear form (p,q)_F = p(partial) q on homogeneous
polynomials; positive definite on real harmonic polynomials, nondegenerate
after complexification. Used to show the round orbit of z_1^l spans V_l.
HARMONIC FUNCTION. One with Laplacian zero; equivalently, equal to its
average over tiny spheres.
HOPF CHARGE. The integer eigenvalue of -i D_{J_0} on V_l; on a monomial
of holomorphic degree b and antiholomorphic degree l-b it equals 2b-l.
Its square is the eigenvalue of -D_{J_0}^2.
LIE ALGEBRA AND BRACKET. A real vector space of matrices closed under
[A,B] = AB - BA. "Generates the full sl(V_l)" means that real linear
combinations and nested brackets of the generators fill sl(V_l).
POLYNOMIAL GROWTH OF DEGREE AT MOST k. |u(x)| <= C(1+d_g(o,x))^k at
every point, simultaneously.
RICCATI EQUATION. P_l' = A_l - p P_l - P_l^2, with P_l symmetric. Its
solution P_l is the logarithmic derivative of the value transfer of
center-regular solutions; two-sided barriers keep it bounded, and a
long round rest resets it near theta_l Id.
SPECTRAL DATA. The fixed finite package: odd m = 2s-1 with s >= 4,
cutoff chi, 0 < a_* < 1 < q_*, integers k, L >= 2 and 0 <= M_l <= N_l
obeying the link-margin (3.11) on [1,q_*], the cutoff-smallness (4.28),
the mean bound (4.29), and the surplus (4.30).
SPHERICAL HARMONICS V_l. Restrictions to S^m of homogeneous harmonic
polynomials of degree l in m+1 variables; dim N_l = C(m+l,m) -
C(m+l-2,m); the round Laplacian acts on V_l by B_l = l(l+m-1).
TRANSMISSION. The linear map sending the value vector of a center-regular
solution at the start of a period to its value vector at the end. Over a
tuned period it equals s_{l,j} D_{l,j} Pi_l, with s_{l,j} > 0.
VOLUME-NORMALIZED. Volume is kept independent of the squash parameter q,
so the whole-space volume growth is controlled.
WEYL TENSOR. The trace-free part of the Riemann curvature, whose vanishing
is the condition of local conformal flatness. It is nonzero on our nonround
cone limits, so our metric is not locally conformally flat.