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A CLOSED RICCI FLOW WITH BOUNDED SCALAR CURVATURE
AND FINITE-TIME CURVATURE BLOWUP
version 2. Written to be read straight through by someone who knows
calculus and vectors and has never seen a Riemannian manifold.
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1. THE QUESTION
1.1 Spaces, lengths, curvature
A manifold of dimension n is a space that near each point looks like a piece
of R^n. The sphere is a 2-dimensional manifold. The torus is a 2-dimensional
manifold. The product S^2 x S^3, whose points are pairs (a point of S^2, a
point of S^3), is a 5-dimensional manifold. Closed means compact with no
boundary: the sphere is closed, the plane is not compact, the closed disk has
a boundary.
A metric g on a manifold is a rule giving every tangent vector a length, and
hence giving lengths of curves, angles between curves, and volumes. Two
metrics on the same manifold are two different geometries on the same space. A
Riemannian manifold is a manifold together with a metric.
A geodesic is a curve that is locally the shortest path between its endpoints:
the analogue of a straight line.
Curvature is about what nearby geodesics do.
The sectional curvature K(P) of a 2-plane P at a point: start two geodesics
in the two directions that span P. If they bend toward each other, K(P) is
positive. If they spread apart, K(P) is negative. The unit sphere has
K = 1 on every plane, Euclidean space has K = 0 on every plane.
The Ricci curvature Ric(u) in a unit direction u: add up the sectional
curvatures of the planes spanned by u and each member of an orthonormal
basis of the directions perpendicular to u. It is an average over the
planes through u, counted with multiplicity.
The scalar curvature R at a point: add up the sectional curvatures of all
planes spanned by an orthonormal basis. Equivalently, R is the trace of the
Ricci tensor, R = tr_g Ric. It is one single number attached to each point.
These three carry strictly decreasing amounts of information:
bounded |Rm| implies bounded |Ric| implies bounded |R|,
up to constants depending only on the dimension n. None of the implications
runs backwards. The arithmetic reason is that R is a sum whose terms carry
signs. Take the 2x2 symmetric matrix with N in one diagonal slot and -N in the
other. Its trace is zero for every N. Its size is N, as large as you like. A
bounded sum of signed numbers says nothing whatever about the size of the
individual numbers, and curvature is exactly a collection of signed numbers.
Every single thing in this document is that last sentence, made geometric.
1.2 Ricci flow
Ricci flow is the equation
d/dt g(t) = -2 Ric(g(t)).
It is a heat equation for the metric: curvature is smoothed out the way
temperature evens out in a metal rod. It is parabolic, so by Hamilton's
short-time existence theorem any smooth metric on a closed manifold starts a
smooth flow g(t), 0 <= t < T, unique, for some T > 0.
Two facts define the game.
HAMILTON 1995. If T is finite, the flow extends smoothly past T if and only
if |Rm| stays bounded on [0,T). So "a singularity at time T" means exactly
"full curvature becomes unbounded as t approaches T".
SESUM 2005. If |Ric| stays bounded on [0,T), the flow extends smoothly past T.
Sesum's theorem makes the next question the natural one. Rm is the curvature
tensor; Ric is its trace; R is the trace of Ric. If a bound on the trace of Rm
forces extension, does a bound on the trace of the trace also force it?
THE SCALAR-CURVATURE EXTENSION CONJECTURE (as stated in Bamler's survey,
page 756). Every finite-time singularity of a closed Ricci flow in dimension
at least four has unbounded scalar curvature; equivalently, a uniform bound
on scalar curvature forces smooth extension.
Theorem 1 below destroys this conjecture in sufficiently high dimension. It
says nothing about dimension four, and no dimension-four statement is made or
implied anywhere in this document. What is at stake here is smooth extension on
the original manifold, which is a different conclusion from continuing the
flow through some singular space.
1.3 The two speeds of blowup
Measure curvature in the natural parabolic units. Instead of |Rm|, look at
(T - t) * max over M of |Rm|.
If that stays bounded as t approaches T, the singularity is Type I: curvature
grows exactly like 1/(T-t), the parabolic rate. If it is unbounded, the
singularity is Type II: curvature grows faster than the parabolic rate.
Wang 2012 proved that at a finite singular time T the quantity
(T - t) * sqrt(max over M of |Rm|) * sqrt(max over M of |R|)
has a strictly positive limit superior. Read that in the contrapositive: if |R|
is bounded, then
limsup as t approaches T of (T - t)^2 * max over M of |Rm|
is strictly positive. So a bounded scalar curvature forbids blowup much slower
than 1/(T-t)^2. Note carefully what this is and is not. It is a statement
about a limit superior, i.e. about a lower bound holding infinitely often, not
a lower bound at every sufficiently late time. It tells you what to aim for. A
bounded-scalar singularity, if one exists at all, must be fast and Type II. The
question is only whether such a thing exists.
1.4 What was known before
Angenent and Knopf built rotationally symmetric neckpinches on spheres. Gu and
Zhu built Type II singularities on spheres. Angenent, Isenberg and Knopf built
degenerate neckpinches with prescribed rates, together with a Bryant-soliton
tip model and a shrinking-cylinder model. All of this shows that full curvature
can concentrate faster than the parabolic rate. None of it controls scalar
curvature; that was not what those constructions were for.
Stolarski built closed doubly warped Ricci flows on S^p x S^{q+1} that develop
conical singularities with arbitrarily fast curvature blowup and that approach
a Ricci-flat cone at the parabolic scale. His own discussion points out that
bounded scalar curvature could be a feature of these examples, but the
construction does not prove a scalar bound. The geometric cap models in that
formal discussion belong to the cohomogeneity-one Ricci-flat setting developed
by Bohm.
What is new here, in two sentences. For a suitable choice of his parameters the
scalar curvature is bounded, because of (i) a quantitative cap estimate, which
uses a strict one-sided comparison with the cone plus curvature point-picking
and does not need convergence to any particular complete cap metric, and (ii) a
dimension-explicit Ricci reaction bound, coming from a one-dimensional
parabolic estimate on cylinders that move with the radial drift, which gives
|Ric| <= C r^{-e} with 0 < e < 1; the square of that bound then fits under a
bounded supersolution for scalar curvature. Everything else is bookkeeping:
choosing the large parameters in the correct order so that all the small
quantities are small simultaneously.
2. THE ANSWER
THEOREM 1. There are an integer q >= 10, a time T > 0, and a smooth Ricci flow
g(t), 0 <= t < T, on the closed connected manifold M = S^2 x S^{q+1}, with
smooth initial metric, such that
sup over M x [0,T) of |R| is finite, and
lim as t approaches T of max over M of |Rm| is infinite.
In particular T is the finite maximal existence time of this flow.
The phrase "q >= 10" is a floor on the statement, not a sharp threshold. The
proof needs q large enough for its constants, and the specific numbers used
below need q >= 12. Any larger fixed q does just as well.
COROLLARY 2. For every sufficiently large integer q put
d_q = ( q + 1 - sqrt( (q+1)(q-7) ) ) / 2.
For every sufficiently large even integer k, with the threshold allowed to
depend on q, there is a smooth Ricci flow on S^2 x S^{q+1} with finite maximal
time T, uniformly bounded scalar curvature, and constants
0 < c <= C < infinity such that
c (T-t)^(-2k/d_q) <= max over M of |Rm|(.,t) <= C (T-t)^(-2k/d_q)
for all sufficiently late times. The flow, the time T, and the constants may
depend on q and k.
So in one fixed dimension there are such examples whose power-law exponent
ranges over an unbounded discrete set. Since 2k/d_q > 1 they are Type II, and
since 2k/d_q grows without bound as k grows, the exponent is not bounded above
even with the dimension held fixed. The upper rate is controlled as well as the
lower rate, which is what makes the word "rate" honest.
The exponent 2k/d_q is worth staring at. It is the ratio of two things: k, the
mode index of the construction, which may be taken as large as we like, and
d_q, a fixed number close to 2 once q is large. The blowup rate grows linearly
in the mode index at fixed dimension.
Sample numbers, so that the parameters stop being abstract:
q m = q+2 d_q a - d_q 2k/d_q at k = 12
------------------------------------------------------------------
12 14 2.469 0.031 9.7
20 22 2.238 0.262 10.7
50 52 2.085 0.415 11.5
100 102 2.040 0.460 11.8
1000 1002 2.005 0.495 12.0
Here a = 5/2 is fixed in Section 9, and d_q tends to 2 from above. The quantity
a - d_q must sit between 0 and 1 for the scheme to work, which is exactly what
the table is checking. It also shows why q >= 12 is the real floor: below it,
d_q exceeds 5/2 and no exponent e is available.
What Theorem 1 does not say. It does not produce a four-dimensional example. It
does not classify anything. It does not claim that scalar curvature is a
useless notion, only that it is not sufficient on its own. Scalar curvature
still detects plenty of singularities; it just does not detect all of them.
3. THE SHAPE OF THE EXAMPLE
3.1 A metric with two sphere factors
The flows in Theorem 1 all have the same shape. Away from two special orbits
called the poles, the metric is
g = ds^2 + phi(s,t)^2 g_{S^2} + r(s,t)^2 g_{S^q}, p = 2,
where s is arclength in the metric g at the time in question, phi and r are
positive smooth functions of (s,t) on each open hemisphere, and g_{S^2},
g_{S^q} are the round unit sphere metrics, the ones with sectional curvature
one. This is called a doubly warped product, and the whole flow lives in this
ansatz, which is why the problem reduces to two unknown functions of two
variables instead of a tensor equation.
Picture a long sock. At each end, the S^q family of spheres shrinks to a
point, so the end is a single S^2 orbit of radius phi(0,t); in the middle the
space looks like S^2 x S^q. The two ends are mirror images. Concretely:
the flow is invariant under the reflection exchanging the two poles;
on each open hemisphere, oriented from its pole toward the equator, r_s > 0;
at a pole, r = 0, r_s = 1 and phi_s = 0, and the usual polar expansions
hold: r is odd in the distance from the pole, phi is even;
phi > 0 at every finite time, so the S^2 orbit at a pole never collapses.
The manifold is S^2 x S^{q+1}, of dimension p + q + 1 = q + 3. The roles of the
two sphere factors are not symmetric: the S^q factor collapses at the poles,
the S^2 factor does not.
3.2 Where and why the curvature blows up
The mechanism of the singularity is easy to state. At a pole, phi_s = 0, so the
sectional curvature of a 2-plane tangent to the S^2 orbit is
j(0,t) = (1 - phi_s^2)/phi^2 = 1/phi(0,t)^2.
If phi(0,t) tends to zero, that sectional curvature tends to infinity and full
curvature blows up. Nothing else has to happen. Section 6 proves the matching
upper bound, which says phi(0,t) is comparable to theta, and theta tends to
zero, so the blowup is not merely possible but forced and rate-controlled.
Now the actual difficulty. The very same shrinking S^2 orbit also pushes
individual components of the Ricci tensor to infinity, and there is no
general reason for the trace to survive. The honest accounting is the formula
r^2 R = p(p-1)(r/phi)^2 + q(q-1)(1-z)
- p z [ (p+1) f^2 + (2q-2) f + 2 r f_r ]
- (p f + q) r z_r,
where u = log phi, f = r u_r, z = r_s^2, and f_r is the derivative of f with
respect to r. Every one of these terms has a geometric meaning, and every one
is computed directly from the warping equations in Section 5. At the cap scale
r is comparable to theta and phi is comparable to theta, so the first term is
bounded, not large, and the remaining terms are bounded because z is pinned
near B^2. The balance that keeps R finite is a real balance between the radial
sectional curvature and the sphere curvature near the pole, not an accident of
notation, and the point of Sections 5 to 9 is to prove it holds uniformly as
t approaches T.
There is one piece of genuinely free good news. Scalar curvature obeys
(d/dt - Laplacian) R = 2 |Ric|^2 >= 0,
so by the maximum principle R can never drop below its initial minimum. The
lower bound in Theorem 1 is free, valid on any closed Ricci flow. All the work
is in the upper bound.
3.3 The two scales
Set
delta = T - t, tau = -log delta, theta = delta^sigma,
where sigma = k/d is a fixed number slightly larger than 1/2, so that
theta << sqrt(delta) at late times. These are the two scales:
the parabolic scale sqrt(delta), the scale at which a singularity is
"expected" to look self-similar;
the cap scale theta = delta^sigma, strictly smaller, the scale at which the
pole orbit lives.
They do different jobs. Rescale the metric by delta^{-1} and look on a fixed
annulus a_1 sqrt(delta) <= r <= a_2 sqrt(delta) with fixed positive a_1 < a_2.
There the rescaled metric is C^2-close to the Ricci-flat cone whose link is
A^2 g_{S^2} + B^2 g_{S^q}, with an error of order delta^nu. Curvature depends
smoothly on a positive metric and its first two derivatives, so
|Ric| <= C delta^{nu-1} on such an annulus,
which is small. That is the good news, and it is exactly what the conical
construction was designed to give.
The catch is that a fixed annulus with fixed a_1 > 0 does not reach the region
where the singularity lives: theta/sqrt(delta) = delta^{sigma-1/2} tends to
zero, so r comparable to theta is far below every fixed annulus at late times.
Small Ricci curvature on the cone scale does not control the cap. That is the
first new estimate, in Section 6.
3.4 The three facts about the geometry at the cap
Everything else rests on these three statements, proved in Sections 5 to 7.
RADIUS SCALE. r^2 |Rm| <= C. Full curvature is controlled once you are at
distance at least r from the pole. In particular the S^2 orbit radius phi is
bounded below away from the pole, and the warping functions cannot develop
wild slopes.
POLE SIZE. c theta <= phi(0,t) <= C theta. The pole orbit shrinks at
exactly the cap rate, no faster and no slower. The upper bound is the
singularity; the lower bound is what stops the curvature from blowing up
faster than the geometry can support.
CAP SLOPE. 0 <= 1 - r_s <= C r/theta for 0 <= r <= theta. Near the pole
the radial slope r_s approaches 1 at a controlled rate, so the metric near
the pole is metrically close to a cone even at the smallest scale.
With those in hand, Section 6 shows (r^2 + theta^2) |Rm| <= C, which is a full
curvature bound at both scales at once. Then Sections 8 and 9 turn a full
curvature bound with good dimension dependence into a Ricci bound, and a Ricci
bound into a scalar bound.
4. THE INPUT FROM STOLARSKI
4.1 The parameters
Set p = 2 throughout and write
m = p + q, A^2 = (p-1)/(m-1), B^2 = (q-1)/(m-1),
d = ( m-1 - sqrt((m-1)(m-9)) ) / 2,
nu = k - d/2, sigma = k/d,
delta = T - t, tau = -log delta, theta = delta^sigma,
gamma = r / sqrt(delta).
Here q >= 10 and k is a sufficiently large even positive integer. Since d is
slightly above 2 for moderate q, nu = k - d/2 is positive for all the k we will
use. Two identities link the scales and are used constantly:
nu/d = sigma - 1/2, and sqrt(delta) * delta^(nu/d) = theta.
The first says the two exponents are not independent. The second says: the
interface radius of the profile region, which is Y_1 delta^(nu/d) in the gamma
variable, equals Y_1 theta in the r variable.
Why A and B take those values. The product of a round sphere of radius A with a
round sphere of radius B has Ricci tensor (p-1) g_{S^p} + (q-1) g_{S^q} in the
two factors, which is the same as (m-1) times the metric exactly when
A^2 = (p-1)/(m-1) and B^2 = (q-1)/(m-1). With those values the link
A^2 g_{S^2} + B^2 g_{S^q} is Einstein with constant m-1, and therefore the cone
over it is Ricci-flat. That cone is the model geometry at the parabolic scale,
and the whole point of the construction is to approach it while the pole orbit
shrinks faster.
Note the useful identities
1 - B^2 = 2/(q+1), and z = r_s^2 has equilibrium value B^2.
So "the slope r_s is close to B" and "the cone model is accurate" are the same
statement, which is why the profile bounds below are stated for z - B^2.
4.2 The proposition
PROPOSITION 4.1 (construction input). For the parameters above, Stolarski's
construction supplies a smooth Ricci flow up to a finite time T on
S^p x S^{q+1} with the following properties, after translating its starting
time to zero.
(i) Off the two pole orbits it has the form g = ds^2 + phi^2 g_{S^p}
+ r^2 g_{S^q}, with s radial arclength at each time, invariant under
reflection exchanging the poles. On each open hemisphere, oriented from
its pole toward the equator, r_s > 0. At a pole r = 0, r_s = 1, phi_s = 0,
the usual odd and even polar expansions hold for r and phi, and phi > 0
at every t < T.
(ii) Write u = log phi, z = r_s^2,
Ut = log( phi / ((A/B) r) ), Zt = z - B^2.
There are positive tolerances eta_j^U, eta_j^Z, large fixed numbers
1 < Y_1 <= Y_2, constants M_0 > 0 and 0 < beta < 1/2 such that, for
j = 0, 1, 2,
| d^j/dgamma^j ( Ut - delta^nu U_k ) |
<= eta_j^U delta^nu ( gamma^(-d-j) + gamma^(2k-d) ),
| d^j/dgamma^j ( Zt - delta^nu Z_k ) |
<= eta_j^Z delta^nu ( gamma^(-d-j) + gamma^(2k-d) ).
The first holds from gamma = Y_1 delta^(nu/d) out to
gamma = M_0 exp(beta tau); the second from gamma = Y_2 delta^(nu/d) out to
the same upper endpoint; both within the hemisphere. The profiles are
smooth on gamma > 0. The function U_k is gamma^(-d) times a polynomial of
degree k in gamma^2, with positive leading coefficients at both ends:
U_k(gamma) ~ c_1 gamma^(-d) as gamma -> 0 and
U_k(gamma) ~ c_2 gamma^(2k-d) as gamma -> infinity, c_1, c_2 > 0.
Also Z_k(gamma) = O(gamma^(-d)) as gamma -> 0.
(iii) For all sufficiently late times, with f = r u_r,
phi/r >= A/B, 0 <= f <= 1, f_r >= 0 for 0 < r <= Y_1 theta,
B^2 <= z <= 1 for 0 < r <= Y_2 theta.
These inner intervals lie strictly below the equator.
(iv) At the initial time tau_0 = -log T the same weighted profile bounds hold
up to an initial outer cutoff G_init = M_0 exp(beta_init tau_0), with
0 < beta_init < 1/2, and Ut >= 0 beyond that cutoff. The exponents lie in
the ranges permitted by the construction; in particular one may take
0 < beta < nu/(2 nu + 1) <= beta_init < 1/2.
They are therefore distinct choices, not two arbitrary exponents. The
tolerances may be chosen small depending on q and k. The inner cutoffs can
then be increased in order, first Y_1 and then Y_2, and afterwards the
initial rescaled time can be increased; each increase is subject only to
lower-size requirements from the construction.
4.3 The profile bounds in plain terms
Ignore the derivatives for a moment and read item (ii) as a single claim:
Ut is delta^nu U_k plus an error of order delta^nu times
( gamma^(-d) + gamma^(2k-d) ).
So on a window of gamma values the geometry is delta^nu close to the cone, and
the discrepancy is described by an explicit profile U_k. The error has two
pieces. The piece gamma^(-d) blows up as gamma goes to zero, which is why the
window needs a lower cutoff, at gamma comparable to Y_1 delta^(nu/d). The piece
gamma^(2k-d) blows up as gamma goes to infinity, which is why the window needs
an upper cutoff; the upper cutoff is not a constant but
M_0 exp(beta tau), which grows like delta^(-beta), and since beta < 1/2 the
corresponding radius M_0 delta^(1/2-beta) still tends to infinity. So the
window is enormous: it covers everything from the cap scale out to enormous
radius. Translated back to r, the window is
Y_1 theta <= r <= M_0 delta^(1/2 - beta),
and everything between the cap and the far exterior is under profile control.
The nuisance is the term delta^nu U_k itself, which is not small. What saves
the day is that it is the same function at every time, up to the delta^nu
factor, so its effect on curvature is explicit and can be bounded once and for
all. That is why U_k's two asymptotics, one at each end, are stated as part of
the input.
Item (iii) is the inner region, below Y_1 theta, where the profiles are not
available. There the assumptions are inequalities instead: the metric is on the
far side of the cone in the ratio phi/r, the logarithmic slope f lies in [0,1]
and is monotone, and z is trapped in [B^2,1]. These are exactly what the pole
size lemma needs. The intervals in (iii) lie strictly below the equator, which
is a sanity condition: at the equator z = 0, and B^2 > 0, so z in [B^2,1]
cannot hold there.
4.4 Where the proposition comes from
The existence statement is the solution furnished by Lemma 3.12 and the proof
of Theorem 1.1 of Stolarski's paper. Assembly, source by source:
weighted profile estimates Definition 3.1;
the initial family Definition 3.5 and Lemma 3.6, with the outer
conditions of Definition 3.2;
the inner inequalities Inner Region Barriers I of Definition 3.3, as
propagated by Lemmas 4.5 to 4.9 and explicitly
invoked in the proof of Theorem 4.3 on page 25
of that paper;
eigenprofile properties Propositions A.12, A.13 and A.21;
the parameter order Appendix B.
One warning about that list. The proof of Theorem 4.3 does use the barrier
conditions on the inner intervals, even though the shorter list in Definition
4.1 of the final class P does not mention all of them. The item (iii) list above
is the list that is actually in force.
Why the initial weighted bounds hold up to the initial expansion's own upper
endpoint, and not merely to the smaller propagated endpoint: the lower U-modes
in Definition 3.5 are gamma^(-d) times polynomials in gamma^2 of degree below
k, and by Propositions A.10 and A.19 the homogeneous lower Z-modes are gamma^2
times polynomials with growth exponents strictly below 2k - d. There are only
finitely many such modes, and each one, together with its first two
derivatives, is bounded by the corresponding weight in the displayed
inequalities on all of gamma > 0. Taking the lower-mode coefficient bound at
the initial time to be eps_0 delta^nu with eps_0 small enough gives the needed
tolerances. This choice is compatible with the degree argument in Lemma 3.10:
the projection errors in Lemmas 6.13 and 6.14 can be reduced afterwards by
increasing the initial rescaled time. Definition 3.5 imposes the outer
conditions starting at that same initial expansion endpoint, so the initial
profile bounds and the outer sign condition meet exactly at G_init, and no
identification of beta_init with beta is needed. The relation between their
permitted ranges is specified in the proof of Theorem 4.3, page 26, of that
paper.
4.5 Translation of notation
The source's notation and the notation of this document are related by
B^2 lambda_k = -nu, alpha_k = nu/d, Y_1 = Upsilon_U,
Y_2 = Upsilon_Z, sqrt(delta) Y_i exp(-alpha_k tau) = Y_i theta.
Thus the source's moving cutoff, which is a constant times exp(-alpha_k tau)
in the gamma variable, becomes a constant times theta in the r variable. The
log-radius convexity condition in the source becomes the slope inequality in
item (iii) through
f = 1 + gamma Ut_gamma,
f_r = (gamma/sqrt(delta)) ( Ut_(gamma gamma) + Ut_gamma / gamma ).
The mode index k used here is the index inside the construction. It is not the
arbitrary curvature blowup exponent appearing in the statement of that paper's
Theorem 1.1, and the two should not be confused. Here k does double duty, once
as the mode index and once as the exponent 2k/d controlling the curvature
rate, and that coincidence is the point of the whole construction.
4.6 What is deliberately not imported
This must be said clearly, because it is the difference between citing a paper
and assuming a theorem. Proposition 4.1 is the entire external existence input.
We do not import:
a scalar curvature upper bound;
a full curvature bound at the cap scale;
convergence to any particular complete cap metric.
Everything in that list is proved in Sections 5 to 9. The cap estimate there
uses only the quantitative inputs stated in Proposition 4.1.
5. GEOMETRIC IDENTITIES AND THEIR CONSEQUENCES
5.1 The warping equations
Everything is computed from the ansatz, so nothing here needs a general
curvature formula. Start with a radial coordinate x fixed on the manifold so
that ds = chi(x,t) dx, and let a subscript s mean differentiation with respect
to arclength at the fixed time under consideration. On a regular orbit the
sectional curvatures of the five types of 2-plane are
h = -phi_ss/phi radial with an S^2 direction,
h_q = -r_ss/r radial with an S^q direction,
j = (1 - phi_s^2)/phi^2 two S^2 tangent directions,
l = (1 - r_s^2)/r^2 two S^q tangent directions,
mu = -phi_s r_s/(phi r) one S^2 and one S^q direction.
The pattern is worth memorizing: a tangent-tangent curvature is
(1 - f_s^2)/f^2 and a radial-tangent curvature is -f_ss/f, for a warping
function f. Two checks. Take f = s, which gives the flat metric
ds^2 + s^2 g_{S^n} in polar coordinates: then f_s = 1 and f_ss = 0, so both
curvatures vanish, correctly. Take the round sphere of radius rho, where
f = rho sin(s/rho): then f_s = cos(s/rho) and f_ss = -f/rho^2, so
(1 - f_s^2)/f^2 = 1/rho^2 and -f_ss/f = 1/rho^2, again both correct.
The Ricci eigenvalues in the radial direction and in each fiber direction are
lambda_0 = p h + q h_q,
lambda_p = h + (p-1) j + q mu,
lambda_q = h_q + (q-1) l + p mu,
and the scalar curvature is R = lambda_0 + p lambda_p + q lambda_q. Notice the
structure already: R is a signed sum of the quantities above, and the single
potentially huge term can sit in lambda_p while being cancelled inside R.
The flow equations, all in the fixed x coordinate, are
d_t r = r_ss + (q-1)(r_s^2 - 1)/r + p phi_s r_s/phi,
d_t phi = phi_ss + (p-1)(phi_s^2 - 1)/phi + q r_s phi_s/r,
[d_t, d_s] = lambda_0 d_s.
These come straight from writing out -2 Ric in this ansatz; they are the usual
doubly warped Ricci flow equations. The third line is the commutator of time
differentiation and arclength differentiation. It is not zero, because
arclength is measured with a metric that itself changes with time. That
commutator is the reason the choice of radial coordinate matters, and it is
what makes the "moving cylinder" trick in Section 7 necessary rather than
merely elegant.
5.2 Three global bounds
LEMMA 5.1 (global elementary bounds). For each flow in Proposition 4.1 there is
a time-independent constant C such that
r <= C, |r_s| + |phi_s| <= C,
and R >= -C throughout the flow. In particular r is uniformly Lipschitz in the
metric g(t).
Proof. At a positive spatial maximum of r, the first derivative vanishes and
the first warping equation gives d_t r <= -(q-1)/r < 0, so a maximum can only
decrease, and hence r is bounded by its initial maximum. For the slope,
differentiate the first equation, commuting time and arclength derivatives
with the commutator above, to get
d_t r_s = (r_s)_ss + ( p phi_s/phi + (q-2) r_s/r ) (r_s)_s
+ [ (q-1)(1 - r_s^2)/r^2 - p phi_s^2/phi^2 ] r_s.
At a positive interior maximum with r_s > 1 the reaction is negative, and the
converse holds at a negative minimum below -1. So |r_s| is bounded by the
larger of one and its initial supremum. Interchanging (r,q) with (phi,p) gives
the same for |phi_s|. On the full interval between the poles the endpoint
values are r_s = 1, -1 and phi_s = 0, so any offending extremum is interior and
the argument applies on a compact time slab without extra boundary conditions;
the singular-looking radial coefficients at the endpoints are harmless for the
same reason. Finally (d_t - Laplacian)R = 2|Ric|^2 >= 0, and the maximum
principle on a closed manifold gives R at time t at least the initial minimum.
5.3 One coordinate choice that makes everything work
Where r_s > 0, use r itself as the spatial coordinate, and regard
u = log phi, v = r_s = sqrt z
as functions of (r,t). Time derivatives from now on fix r, not x. Subtracting
the advection term (d_t r) d_r from the warping equations gives the sideways
form:
u_t = z u_rr + ((q-1+z)/r) u_r - (p-1) e^{-2u},
v_t = z v_rr + ((q-1-z)/r) v_r + (q-1) v (1 - v^2)/r^2 - p v^3 u_r^2.
The corresponding geometric heat operator on a function of (r,t) is
D F := (d_t - Laplacian) F(r,t)
= F_t (at fixed r) - z F_rr - ((q-1+z)/r) F_r.
The derivation is one line and worth doing once. Since r is a radial function,
Laplacian r = r_ss + q r_s^2/r + p phi_s r_s/phi = r_ss + q z/r + p z u_r,
while the flow equation gives d_t r = r_ss + (q-1)(z-1)/r + p z u_r. Subtract:
d_t r - Laplacian r = (q-1)(z-1)/r - q z/r = -(q-1+z)/r. Then for any F,
D F = F_t (fixed r) - z F_rr + F_r ( d_t r - Laplacian r ),
which is the displayed formula. This is a composition identity for the
geometric operator, so it stays valid across the equator even though the (r,t)
coordinates themselves degenerate there.
Two consequences of this form are used over and over.
The logarithmic slope equation. Applying D to log phi and to log r gives
D log(phi/r) = (q-1)/r^2 - (p-1)/phi^2.
Its right side is a difference of two positive numbers with a single zero,
at phi^2/r^2 = (p-1)/(q-1) = A^2/B^2, that is at phi/r = A/B. And the sign
is stabilizing: below the cone ratio the second term dominates and the
reaction is negative, which pushes log(phi/r) up, and above the cone ratio
the reaction is positive, which pushes it down. The cone ratio A/B is a
stable equilibrium of a one-dimensional heat equation with an explicit
reaction. That single observation is the engine behind both the strict inner
gap and the exterior Ricci bound.
Power functions have explicit D. For any real exponent alpha,
D ( r^alpha ) = -alpha (q-1 + alpha z) r^(alpha-2), r > 0.
So on a bounded-z region, applying D to r^alpha produces a definite sign
times r^(alpha-2). That is how the barriers in Section 8 are built and
checked: choose the exponent, look at the sign.
5.4 The strict inner gap
LEMMA 5.2 (a strict inner gap). After choosing the tolerances small enough,
there are constants eps_1 > 0 and C < infinity such that, at all sufficiently
late times,
phi/r >= A/B + eps_1 for 0 < r <= Y_1 theta,
phi(Y_1 theta, t) <= C theta.
Proof. At the interface gamma_1 = Y_1 delta^(nu/d) we have
delta^nu gamma_1^(-d) = Y_1^(-d) exactly, while the relative error in the
leading small-gamma asymptotic of U_k tends to zero. Choose
eta_0^U < c_1/4. Then the profile bounds in Proposition 4.1 give
0 < (1/2) c_1 Y_1^(-d) <= Ut(gamma_1, tau) <= C_{q,k,Y_1}
at late times, which proves both assertions at the interface. Now
d/d(log r) log(phi/r) = f - 1 <= 0 by item (iii), so log(phi/r) is
nonincreasing as r increases; the lower bound at the interface therefore
propagates inward to all smaller r. Any positive number below
(A/B)(exp(c_1 Y_1^(-d)/2) - 1) is an admissible eps_1.
The logic of that proof is worth pausing on, since it is the pattern used
everywhere. The profile estimate pins the geometry down at one radius. The
monotonicity f <= 1 then carries the information inward. Nothing has to be
known at the pole itself.
5.5 Uniform bounds used to choose the dimension
LEMMA 5.3 (bounds used in the dimension estimate). The construction parameters
can be chosen so that, for all sufficiently large q and all sufficiently late
times,
|z - 1| <= C_0/q, r |u_r| <= 2,
(p-1) r^2 e^{-2u} <= (q-1)(1 + C_0/q)
on 0 < r <= sqrt(delta), where C_0 is a numerical constant independent of q
and k.
Proof. For 0 < gamma <= 1 the fixed profiles and their errors satisfy
|Ut| + gamma|Ut_gamma| <= C_{q,k} delta^nu gamma^(-d) and
|Zt| <= C'_{q,k} delta^nu gamma^(-d) above their respective lower cutoffs.
Enlarge Y_1 and then Y_2 until the right sides at those cutoffs are at most
1/q; they only decrease as gamma increases, so this suffices for the whole
range. In the remaining inner range use the inequalities of item (iii). Now
use the three identities
1 - B^2 = 2/(q+1), r u_r = 1 + gamma Ut_gamma,
(p-1) r^2 e^{-2u} = (q-1) e^{-2 Ut}.
They give |z - 1| <= 3/q, r|u_r| <= 1 + 1/q, and the third bound with
e^{2/q} <= 1 + 4/q for large q, so C_0 = 4 works. All the dimension- and
mode-dependent profile constants have been absorbed into the choice of
cutoffs, which is exactly why C_0 does not depend on q or k.
One point needs justifying and is easy to miss: the interval
0 < r <= sqrt(delta) is said to be below the equator. Why is the equator not
inside it? Because the lower cutoffs tend to zero and the upper cutoff exceeds
one at late times, so the profile region reaches past gamma = 1. If the equator
occurred before gamma = 1, the same estimate run up to that endpoint would
contradict its value z = 0, by continuity. So the stated interval really is
available in the r coordinate.
5.6 A global lower ratio, and small Ricci on annuli
LEMMA 5.4 (global ratio and parabolic annuli). For a flow chosen as above,
phi/r >= c_0 > 0 globally off the poles. Also, for every fixed
0 < a_1 < a_2 < infinity, at all sufficiently late times,
|Ric| <= C_{a_1,a_2} delta^(nu - 1)
for a_1 sqrt(delta) <= r <= a_2 sqrt(delta).
The constants and the required starting time may depend on the annulus.
Proof of the ratio bound. The positive large-gamma coefficient of U_k, together
with a tolerance small compared with that coefficient, gives Ut >= 0 for all
gamma >= Gamma_0 in the profile region, where Gamma_0 is a fixed sufficiently
large number. At the initial time use the initial profile bounds up to
G_init and the outer sign condition beyond that same endpoint. Increase the
construction's starting rescaled time so that Gamma_0 lies in the controlled
region at every later time. Then on the whole exterior
{r >= Gamma_0 sqrt(delta)} compare log(phi/r) with log(A/B) using
D log(phi/r) = (q-1)/r^2 - (p-1)/phi^2,
for which log(A/B) is an exact equilibrium. Initial data and moving lateral
data have the right sign, the reaction is locally Lipschitz on each compact
time slab, so comparison applies. Both hemispheres are included and the equator
is not a boundary. On 1 <= gamma <= Gamma_0 profile control bounds Ut from
below, and on gamma <= 1 use Lemma 5.3. On earlier compact time intervals a
positive lower ratio holds by smoothness, positivity of phi, and boundedness
of r.
Proof of the annulus bound. On a fixed positive compact interval of gamma
values the pulled-back metric delta^(-1) g is
dgamma^2/z + (A/B)^2 gamma^2 e^{2Ut} g_{S^2} + gamma^2 g_{S^q}.
It differs in C^2 by O(delta^nu) from the metric with z = B^2 and Ut = 0, and
that limit metric is the Ricci-flat cone with link A^2 g_{S^2} + B^2 g_{S^q}.
Curvature is a smooth function of a positive metric and its first two
derivatives on a fixed annulus, so its Ricci norm is O(delta^nu) in the
rescaled metric; scaling back by delta^(-1), which multiplies curvature by
delta, gives the stated bound. Any prescribed fixed annulus eventually lies
inside the profile region, and positivity of z there excludes the equatorial
endpoint, as in Lemma 5.3.
5.7 Summary of Section 5
Four facts are now available.
r and both warping slopes are globally bounded, and R >= -C.
The operator D has the clean form above, its action on powers r^alpha is
explicit, and log(phi/r) solves a one-dimensional reaction equation whose
equilibrium is exactly the cone ratio A/B.
The cone gap is strict on the inner region, at scale theta.
The ratio phi/r is bounded below globally, and Ricci curvature is small,
of size delta^(nu-1), on any fixed parabolic annulus.
What is still missing is any control at radius comparable to theta, and any
dimension-independent bound on Ricci curvature. Those are Sections 6 and 7.
6. CURVATURE AT THE RADIUS SCALE AND AT THE CAP SCALE
6.1 What Shi's estimate gives, and what point picking does
Two tools are imported in this section and nowhere else.
SHI's derivative estimate (Shi 1989, with precise statements in Cao-Zhu,
Theorems 1.4.1 and 1.4.2). If |Rm| is bounded by K on a parabolic
neighborhood, then |grad Rm| is bounded on a smaller parabolic neighborhood
by a constant depending on the dimension, on K, on the spatial margin, and
on the elapsed time. No lower bound on the injectivity radius is needed.
The point-picking, or record point, argument. There is one pattern, used
twice below. Suppose the quantity you want to bound is not bounded. Take
points (x_i, t_i) with t_i tending to T where it attains a record maximum,
rescale distances by the square root of the curvature there so that the
curvature at the record point becomes 1, move that point to the origin and
its time to 0, and then look at the limit. The record property says that in
the rescaled picture the curvature is at most 1 on everything in the past;
Shi upgrades that to a two-sided bound on derivatives; you are left with an
explicit geometric contradiction. This is how a bound that "should" hold is
turned into a bound that does hold, without any compactness theory of the
whole flow.
The first application controls full curvature at the radius scale.
LEMMA 6.1 (radius scale). There is a time-independent constant C such that
r^2 |Rm| <= C.
Proof. Suppose not. Choose record points (x_i,t_i) with t_i tending to T, put
r_i = r(x_i,t_i) > 0 and Q_i = |Rm|(x_i,t_i), so that
r_i^2 Q_i = max over M x [0,t_i] of r^2 |Rm| tends to infinity.
Such points exist by compactness on each closed time slab. Boundedness of r
(Lemma 5.1) forces Q_i to infinity. By the S^q component of the metric
evolution,
|d_t (r^2)| <= 2 |lambda_q| r^2 <= C_m r^2 |Rm|,
and this identity extends to the poles because r^2 is smooth there. So for a
fixed sufficiently small b > 0 the record bound gives
| r(x,t)^2 - r(x,t_i)^2 | <= r_i^2/16 for t_i - b/Q_i <= t <= t_i,
at every fixed point x of the manifold. These time intervals lie in the flow
for all large i.
Now rescale by Q_i and translate t_i to time 0. On a fixed ball of radius R
about x_i, measured in the rescaled metric at time -b, three things hold: the
Lipschitz bound for r, the displayed time-variation bound, and
r_i sqrt(Q_i) tending to infinity together imply r >= r_i/2 throughout
[-b,0], for large i. The record bound gives |Rm|/Q_i <= 4 on that fixed
spatial domain throughout the interval. Integrating the metric evolution makes
the metrics uniformly comparable there, with a length-comparison factor E that
is independent of both R and i. Choose R > E + 2. Then a final-time curve of
length at most one cannot leave the initial ball: up to the first exit its
initial length would be at least R, hence its final length at least R/E > 1.
So the final unit ball lies inside the initial ball, with a fixed positive
margin measured in the initial metric. Take b below the dimension-dependent
time threshold in the local derivative estimate. That estimate then gives a
uniform bound for the final rescaled |grad Rm| on the ball.
That bound is now contradicted geometrically. In the final rescaled metric the
warping functions are F_1 = sqrt(Q_i) phi and F_2 = sqrt(Q_i) r. Both tend to
infinity at the center, by Lemma 5.4. Their derivatives with respect to
rescaled arclength are the original phi_s and r_s, hence uniformly bounded by
Lemma 5.1. So both functions tend uniformly to infinity on any fixed short
radial interval about the center. Such intervals exist: the Lipschitz bound
makes the rescaled distance to either pole tend to infinity, so a short radial
interval about the center stays away from both poles, and crossing the equator
presents no obstruction.
Meanwhile the rescaled tangential curvatures j, l, mu tend to zero, because
they are homogeneous of degree minus two in the rescaled metric while their
numerator stays bounded. Since the rescaled curvature norm is one at the
center, the identity
|Rm|^2 = 4 ( p h^2 + q h_q^2 + (p(p-1)/2) j^2
+ (q(q-1)/2) l^2 + p q mu^2 )
forces at least one radial curvature -F_a''/F_a to have absolute value bounded
below by a positive dimension-dependent constant. Along a radial geodesic the
radial vector and a fixed sphere-tangent vector divided by its warping factor
are parallel, so the derivative estimate keeps that same sectional curvature
of one sign and bounded away from zero on a fixed shorter interval. On that
interval F_a tends uniformly to infinity. Integrating F_a'' then forces an
unbounded change in F_a', contradicting the slope bound. So the assumed
unboundedness was impossible.
A reading note. The rescaling is by Q_i, the full curvature, but the conclusion
is a two-scale statement involving r and theta. That is why the lemma is only
one half of Proposition 6.4 below: r^2 |Rm| bounded means full curvature is
controlled away from the pole, and says nothing at r comparable to theta.
6.2 How big is the pole orbit
The next lemma is where the scalar curvature sign is actually used, and it is
the quantitative version of an argument of Stolarski's. He shows that an
ancient limit with nonnegative scalar curvature cannot occur. Here we use the
much weaker lower bound R >= -C on the given flow, and integrate over a finite
annulus whose logarithmic width would diverge if the pole orbit were too small.
LEMMA 6.2 (size of the pole orbit). At all sufficiently late times,
c theta <= phi(0,t) <= C theta.
Moreover phi(r,t) >= c theta for 0 <= r <= Y_1 theta.
Proof. The upper bound follows from Lemma 5.2 together with f = r u_r >= 0,
which makes phi nondecreasing in r.
For the lower bound the tool is the scalar curvature written in terms of the
warping functions. Using h, h_q, j, l, mu above and u_r = f/r, z = r_s^2, a
direct computation gives
r^2 R = p(p-1)(r/phi)^2 + q(q-1)(1-z)
- p z [ (p+1) f^2 + (2q-2) f + 2 r f_r ]
- (p f + q) r z_r.
Here is the derivation in one line, since it is used again: r^2 h = -r^2
phi_ss/phi, and phi_ss/phi = z (u_rr + u_r^2) + z_r u_r/2 while r_ss = z_r/2;
substituting u_r = f/r gives the formula.
Suppose the lower bound failed. Take times tending to T for which
L_0 = phi(0,t) = o(theta). The inequalities r_s >= B and |phi_s| <= C give
phi(L_0,t)/L_0 <= 1 + C/B.
Let r_* = (L_0 theta)^(1/2). Since d/d(log r) log(phi/r) = f - 1, the lower
ratio bound of Lemma 5.4 and the upper bound just displayed give
0 <= integral from L_0 to r_* of (1-f) dr/r <= C.
The integrand is nonnegative and nonincreasing, because 0 <= f <= 1 and
f_r >= 0. So
0 <= 1 - f(r_*,t) <= C / log(r_*/L_0) which tends to 0.
Consequently f = 1 + o(1) uniformly on [r_*, theta].
Now freeze f at 1 and look at the first two lines of the scalar formula. The
undifferentiated part before the z_r term becomes
p(p-1)/(phi/r)^2 + q(q-1) - m(m-1) z,
which vanishes exactly at phi/r = A/B and z = B^2. By the strict gap of Lemma
5.2 (which gives phi/r at least A/B + eps_1 on the whole inner region) and by
z >= B^2, that expression is at most -kappa for some fixed kappa > 0. The term
involving f_r is nonpositive, and the uniform error f - 1 = o(1) is harmless
once the flow and the dimension are fixed. Using R >= -C from Lemma 5.1 we get,
after reducing kappa if necessary,
(p f + q) r z_r <= -kappa + C theta^2 <= -kappa/2 on r_* <= r <= theta.
Since p f + q <= m, this gives r z_r <= -kappa/(2m). Integrating contradicts
B^2 <= z <= 1, because log(theta/r_*) = (1/2) log(theta/L_0) tends to infinity.
So the pole lower bound holds. Monotonicity of phi in r extends it from the
pole to the whole indicated inner interval.
The shape of that argument is worth naming, because it is the only place where
the sign of scalar curvature enters. The scalar formula is a sum of a term that
vanishes at the cone ratio and a term proportional to z_r. If the pole were too
small, the annulus across which we integrate would be logarithmically thick,
f would be pinned at 1 across it, the first term would be strictly negative of
fixed size, and z would have to travel further than the interval [B^2,1]
allows. So the pole cannot be too small. This is a one-dimensional integration
argument dressed in geometry.
6.3 The radial slope at the cap scale
LEMMA 6.3 (radial slope at the cap scale). For sufficiently large q, at late
times,
0 <= 1 - v <= C r/theta for 0 <= r <= theta,
where v = r_s.
Proof. Put w = 1 - v. The second sideways equation gives
( d_t - z d_rr - ((q-1-z)/r) d_r + (q-1) v (1+v)/r^2 ) w
= p v^3 u_r^2 <= C theta^(-2).
The last inequality uses u_r = phi_s/(phi v), Lemma 6.2, and B <= v <= 1 at
the cap scale. Apply the same operator to r/theta; the result is exactly
sigma r/(delta theta) + [ (q-1)(v(1+v) - 1) + z ] / (r theta).
For large q, B(1+B) > 1, so this expression is bounded below by a fixed
positive multiple of theta^(-2) when r <= theta. A sufficiently large multiple
of r/theta therefore dominates the source, the data at r = theta, and the data
on any fixed late starting slice.
Comparison at the pole needs one sentence of care. Fix a compact time slab.
Smooth polar expansions give w = O(r^2) uniformly on that slab, so the proposed
upper barrier dominates on a sufficiently small inner boundary r = eta.
Compare on eta <= r <= theta(t) and let eta decrease to zero. The auxiliary
radius may depend on the slab, but the multiplier in the barrier does not. That
proves the display.
6.4 Full curvature at both scales at once
Now the two scales are combined. Recall theta shrinks, so theta^2 |Rm| is a
stronger requirement than r^2 |Rm| near the pole, and r^2 |Rm| is the stronger
requirement away from it.
PROPOSITION 6.4 (full curvature bound). For every flow with the choices made
above, and q sufficiently large, there is a constant C, allowed to depend on
that flow, such that
(r^2 + theta^2) |Rm| <= C
on M x [0,T).
Proof. By Lemma 6.1 it remains to bound theta^2 |Rm|. Suppose it were
unbounded and choose record points with Q_i = |Rm|(x_i,t_i) such that
theta_i^2 Q_i = max over M x [0,t_i] of theta(t)^2 |Rm|(x,t)
tends to infinity, where theta_i = theta(t_i).
Since theta is decreasing, for every t <= t_i,
|Rm|(x,t) <= (theta_i/theta(t))^2 Q_i <= Q_i.
So the global derivative estimate bounds |grad Rm| at the final time after
rescaling by Q_i, uniformly in i. For all large i a fixed backward interval in
rescaled time lies inside the flow.
Now the two scales pull in opposite directions, which is the point. Lemma 6.1
gives r_i sqrt(Q_i) <= C, whereas theta_i sqrt(Q_i) tends to infinity. So any
fixed short outward radial interval from the center lies inside r <= theta_i.
On such an interval the rescaled first warping function satisfies
sqrt(Q_i) phi >= c sqrt(Q_i) theta_i which tends to infinity,
and the rescaled second warping function is bounded above on each such
interval, with arclength derivative v tending to 1 uniformly by Lemma 6.3. If
the center is on a pole orbit, choose any outward radial ray.
Move a fixed sufficiently small positive distance along the ray. The derivative
estimate keeps the curvature norm at least 1/2, while the rescaled radius is
now bounded below by a fixed positive number. At this new center the three
tangential curvatures tend to zero. So by the norm identity of Section 6.1 some
radial curvature has nonzero magnitude and constant sign on a fixed subsequent
interval. For F = sqrt(Q_i) phi, integrating F'' contradicts bounded F' because
F tends to infinity. For F = sqrt(Q_i) r, integrating F'' contradicts the
uniform convergence F' = v tending to 1, because F stays bounded below by a
positive constant on that interval. Both alternatives are impossible, so
theta^2 |Rm| is bounded.
Combining that with Lemma 6.1 gives the display, and the proposition is proved.
Two remarks, because they matter later. First, the constant in Proposition 6.4
may depend on the whole chosen flow, and in particular may grow arbitrarily
fast with the dimension. That is not good enough to choose the dimension, which
is why Section 7 needs a different estimate with explicit dimension dependence.
Second, this is the estimate that does not require convergence to any complete
cap metric. Point picking plus Shi gets the cap bound from the quantitative
inputs alone, which is exactly the claim made in Section 4.6.
7. A RICCI REACTION BOUND WITH EXPLICIT DIMENSION
The constant in Proposition 6.4 may grow arbitrarily with the dimension, which
is useless for choosing the dimension. This section produces a second estimate
whose leading dimension dependence is visible. Everything reduces to a
one-dimensional statement about a single scalar function, which is what keeps
the constants free of q.
7.1 A one-dimensional interior estimate
For R > 0 let P_R = (-R,R) x [-R^2,0], with space coordinate y and time
coordinate rho.
LEMMA 7.1 (heat perturbation). There is eps_0 > 0 with the following property.
If a smooth function H on P_2 satisfies
H_rho = a H_yy + b H_y + F, |a - 1| <= eps_0, |H| + |b| + |F| <= K,
then |H_y| <= C(K) on P_1, including the terminal time by continuity from
below. No bounds on derivatives of the coefficients are required. The same
conclusion holds after any fixed rescaling of the cylinders.
The point of the lemma is exactly the missing hypothesis: the drift b may be
large and may oscillate, as long as it is bounded, and the diffusion coefficient
a need not be smooth. Both features are needed in 7.2, where the coefficients
are built from the flow and have no useful smoothness in the moving frame.
Sketch of proof, with the imported pieces named. Fix an exponent s > 3. The
constant-coefficient heat estimate one uses is
||V_yy||_s + ||V_rho||_s <= C_s ||V_rho - V_yy||_s
for smooth functions compactly supported in space, vanishing in the distant
past, with all norms taken over times up to zero. This is the classical
parabolic Calderon-Zygmund estimate for the heat operator; here the constants
are one-dimensional. It is Theorem 5.5 of Krylov 2006, with time reversed, and
it is equivalent to the L^s boundedness of the second spatial derivative of the
causal heat potential. The half-infinite time version follows by extending the
forcing by zero into the future. Note that the function itself need not vanish
at time zero.
For 1 <= R_1 < R_2 < 2 put d_R = R_2 - R_1 and choose a cutoff zeta equal to
one on P_{R_1} and supported spatially and in the past inside P_{R_2}, with
|zeta_y| <= C d_R^(-1) and |zeta_yy| + |zeta_rho| <= C d_R^(-2). Writing
V = zeta H and expanding directly,
(d_rho - d_yy) V = (a-1) V_yy + b V_y + zeta F
+ (zeta_rho - a zeta_yy - b zeta_y) H
- 2 a zeta_y H_y.
Choose eps_0 so that C_s eps_0 < 1/2 and absorb the first term using the
estimate above. If E(R) denotes the sum of the L^s norms of H_yy and H_rho over
P_R, the remaining terms give
E(R_1) <= C(K) d_R^(-1) ||H_y||_{L^s(P_{R_2})} + C(K) d_R^(-2).
Spatial interpolation on intervals of radius between one and two gives, for
small h > 0 and with no boundary condition on H,
||H_y||_{L^s(P_R)} <= h ||H_yy||_{L^s(P_R)} + C h^(-1) ||H||_{L^s(P_R)}.
Taking h to be a sufficiently small multiple of d_R gives
E(R_1) <= (1/16) E(R_2) + C(K) d_R^(-2).
Iterate with R_j = 7/4 - 2^(-j)/4. The error terms are summable, because their
factors grow like 4^j while the iteration contributes 16^(-j). The remainder
tends to zero because E(R_j) <= E(7/4) < infinity for each individual smooth
function; this finiteness is a property of one smooth function, not a presumed
uniform bound. Hence E(3/2) <= C(K). Interpolation also controls the first
derivative norm, and parabolic Sobolev embedding, at
LSU 1968, Chapter II, Section 3, Lemma 3.3, with s > 1 + 2, bounds the first
spatial derivative on P_1. That is the conclusion.
7.2 Cylinders that move with the radial drift
The reason the cylinder has to move is the commutator in Section 5.1. In the
(r,t) coordinates the coefficient of u_r contains (q-1)/r, which is enormous at
the cap scale; a cylinder centered at fixed radius would leave that coefficient
in the equation and the estimate would lose a factor of sqrt(q). Moving the
center along the radial flow cancels it before the estimate is applied.
LEMMA 7.2 (large-q derivative bounds). For all sufficiently large q and
sufficiently late times,
| r z_r | <= C q^(-1/2), r^2 | u_rr | <= C q^(1/2)
on 0 < r <= sqrt(delta)/8, where C is independent of both q and k.
Proof. Fix (r_c,t_c) in the indicated region and set
h_c = r_c/sqrt(q),
t = t_c + h_c^2 rho,
r = R_c(t) + h_c y,
R_c(t) = sqrt( r_c^2 + 2(q-1)(t_c - t) ).
On P_2 we have 1 <= R_c/r_c <= 3. For q >= 16,
(1/2) r_c <= r <= (7/2) r_c,
r^2 <= (49/256) delta_c < delta(t), delta_c = T - t_c,
and the earliest time in the cylinder is at least t_c - delta_c/(16q). So a
single sufficiently late threshold for t_c, for the chosen flow, puts every
such cylinder inside the region of Lemma 5.3.
Since R_c' = -(q-1)/R_c, the function H = q(v - 1), pulled back to this
cylinder, satisfies Lemma 7.1 with
a = z,
b = h_c [ (q-1)(1/r - 1/R_c) - z/r ],
F = q h_c^2 [ (q-1) v (1-z)/r^2 - p v^3 u_r^2 ].
Check the sizes. |H| <= C_0 because q|v-1| = q|z-1|/(v+1), and |a-1| <= C_0/q.
The apparently large drift is bounded after all:
| h_c (q-1)(1/r - 1/R_c) | = (q-1) h_c^2 |y| / (r R_c) <= C.
For the source use q h_c^2 = r_c^2, (q-1)|1-z| <= C_0, and r|u_r| <= 2. So all
coefficients and amplitudes are bounded by one numerical constant, and
increasing q meets the smallness hypothesis. Lemma 7.1 then gives |H_y| <= C on
P_1, hence
|z_r| = (2v/(q h_c)) |H_y| <= C/(r_c sqrt(q)) on P_1.
For the second estimate, differentiate the first sideways equation with
w = u_r:
w_t = z w_rr + [ z_r + (q-1+z)/r ] w_r
+ [ z_r/r - (q-1+z)/r^2 + 2(p-1)e^(-2u) ] w.
On the smaller cylinder P_1 put W = r_c w. Its transformed drift is
b_W = h_c (q-1)(1/r - 1/R_c) + h_c (z_r + z/r),
and its zeroth-order coefficient is
c_W = h_c^2 [ z_r/r - (q-1+z)/r^2 + 2(p-1)e^(-2u) ].
Lemma 5.3, the inclusion above, and the z_r bound control |W|, |b_W| and |c_W|
numerically; in particular h_c^2 (p-1)e^(-2u) <= C. Treat c_W W as a bounded
source and apply a fixed rescaled version of Lemma 7.1 on P_1. Evaluating at
the center gives
|W_y(0,0)| = r_c h_c |u_rr(r_c,t_c)| <= C,
which is the second estimate. The first follows from the z_r bound at the
center. Only the numerical C_0 and the one-dimensional estimates entered the
constants, so neither q nor k appears in C.
7.3 The Ricci norm and its reaction matrix
The derivative estimates now control the radial sectional curvatures. What
remains is algebra: write the action of the curvature tensor on invariant
diagonal Ricci tensors in coordinates that carry the multiplicities correctly,
and read off the largest eigenvalue.
The tensor evolution, together with the evolution of the metric, gives
(d_t - Laplacian) |Ric|^2 = -2 |grad Ric|^2 + 4 <Rm(Ric), Ric>,
where the inverse-metric derivatives in the squared norm cancel the cubic Ricci
term from the covariant evolution of the tensor. Kato's inequality then gives,
wherever |Ric| > 0,
(d_t - Laplacian) |Ric|
<= (2/|Ric|) <Rm(Ric), Ric>.
For a diagonal tensor the curvature convention in force here is
<Rm(D),D> = sum over i,j of R_ijij D_i D_j, with R_ijij the sectional
curvature of the plane spanned by the i and j directions.
PROPOSITION 7.3 (reaction bounds). At late times on 0 < r <= sqrt(delta)/8 the
reaction quotient
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