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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.
+================================================================================
+
+
+0. HOW TO READ THIS
+
+
+Sections 1 to 3 need only calculus, vectors, and the idea of an average. They
+contain the whole story: what is being asked, why the obvious guess is wrong,
+and what the answer is.
+
+Sections 4 to 9 are the proof. Every step of the argument is stated. Most are
+computations you can check by differentiating. A few use a standard tool from
+analysis -- a heat kernel estimate, a maximum principle, Shi's derivative
+estimate -- and each of those is named at the moment it is used, with the
+precise place where it is proved. When this document says "imported", that is
+exactly what it means: proved elsewhere, cited exactly, not re-proved here.
+That is the only kind of gap in this document, and it is never hidden.
+
+Section 10 says honestly what is new and what is borrowed. Section 11 is a map
+of which lemma feeds which. Section 12 is a glossary of every symbol. Section
+13 is a list of things you can test yourself on.
+
+Two habits make the proof much easier to follow.
+
+ (a) Almost every estimate below has the same shape: "something is at most a
+ constant times a power of r, or a power of delta". Watch the powers. The
+ powers are the plot.
+
+ (b) There are exactly two length scales in the problem, sqrt(delta) and
+ theta, with theta much smaller. Almost every lemma says which of the two
+ it is about. Mixing them up is the main way to get lost.
+
+
+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 How to read the profile bounds
+
+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
+
+# ^C interrupted by user!
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