# A compactness property for solutions of the Ricci flow.

1. In this paper we prove that, given a sequence of complete solutions to the Ricci flow with their curvatures uniformly bounded above and their injectivity radii uniformly bounded below, we can find a convergent subsequence. To make this precise we need a few definitions.To begin with, we fix a time interval A [less than or equal to] t [less than or equal to] [Omega] with -[infinity] [less than or equal to] A [less than] 0 and 0 [less than] [Omega] [less than or equal to] [infinity]. If M is a manifold, we look for a complete time-dependent metric on M with bounded curvature, measuring a smooth one-parameter family of metrics G = G(t) for A [less than] t [less than] [Omega] where each G(t) = {[g.sub.ij](x, t)[dx.sup.i][dx.sup.j]} is a complete metric with Riemannian curvature bounded by a constant B independent of t. A marking on M is a choice of a point Q [element of] M which we call the origin, and an orthonormal frame q at Q at t = 0 with respect to the metric G(0). For simplicity we shall refer to such a collection M = {M, G, Q, q} as an evolving complete marked Riemannian manifold. We say that M is a solution to the Ricci flow if it solves the equation

[Delta]/[Delta]t [g.sub.ij] = 2[R.sub.ij]

for the evolution of the metric by its Ricci tensor.

Definition 1.1. We say that a sequence [M.sub.k] = {[M.sub.k], [G.sub.k], [Q.sub.k], [q.sub.k]} of evolving complete marked Riemannian manifolds converges to the evolving complete marked Riemannian manifold M = {M, G, Q, q} if there exists a sequence of open sets [U.sub.k] in M containing Q and a sequence of diffeomorphisms [F.sub.k] of the sets [U.sub.k] in M to open sets [V.sub.k] in [M.sub.k] mapping Q to [Q.sub.k] and q to [q.sub.k], such that any compact set in M eventually lies in all [U.sub.k] and the pull-backs [G.sub.k] of the metrics [G.sub.k] by the maps [F.sub.k] converge to G on every compact subset of M x (A, [Omega]) uniformly together with all their derivatives.

Because of the choice of an origin and a frame at the origin, it is clear that if the limit exists it is unique up to a unique isometry. If we did not fix the origin, the limit metric might not be unique; for example a sequence of paraboloids with the same shape but the origin moving away can converge to the flat plane. It is also clear that a limit of solutions to the Ricci flow is again a solution to the Ricci flow.

We can now state our precise result.

MAIN THEOREM 1.2. Let [M.sub.k] = {[M.sub.k], [G.sub.k], [Q.sub.k], [q.sub.k]} be a sequence of evolving complete marked Riemannian manifolds which are solutions to the Ricci flow. Suppose that (a) the absolute value of the Riemannian sectional curvatures of the [M.sub.k] at all times A [less than] t [less than] [Omega] are uniformly bounded above by a constant B [less than] [infinity] independent of k, and (b) the injectivity radii of the [M.sub.k] at the origin [Q.sub.k] at time t = 0 are uniformly bounded below by a constant [Delta] [greater than] 0. Then there exists a subsequence which converges to an evolving complete marked Riemannian manifold M = { M, G, Q, q} which is also a solution of the Ricci flow, with its Riemannian sectional curvatures bounded above by B and its injectivity radius at the origin Q at time t = 0 bounded below by [Delta].

2. Here we sketch the main line of argument in the proof. The reader is no doubt aware of the work of Gromov [G], Peters [P1] and [P2], and Green and Wu [G&W], on limits of compact manifolds with bounded curvature, diameter and injectivity radius, and it will suffice to adapt these ideas to our present situation. One thing at least is much easier. The work of W. X. Shi [S] shows that for solutions of the Ricci flow bounds on the curvature automatically produce at subsequent times bounds on all the derivatives of the curvature. We use these in the following way. First it suffices to prove the theorem in the case A [greater than] - [infinity] and [Omega] [less than] [infinity]; for this being done, of A = -[infinity] we take a sequence [A.sub.j] [approaches] - [infinity] and if [Omega] = [infinity] we take a sequence [[Omega].sub.j] [approaches] [infinity], apply the theorem on each finite interval, and by the usual diagonalization trick extract a subsequence converging on the union of the finite interval. Similarly, if we can prove the theorem with the extra assumption of derivative bounds on the curvature, then given a solution on a finite interval A [less than] t [less than] [Omega] for each [[Epsilon].sub.j] [greater than] 0 we get a solution with derivative bounds on the curvature for A + [[Epsilon].sub.j] [less than] t [less than] [Omega] and now letting [[Epsilon].sub.j] [approaches] 0 and finding a subsequence converging on each smaller interval, we find by diagonalization a subsequence converging on all of A [less than] t [less than] [Omega]. Hence we may add to the hypotheses of Theorem 1.1 the following.

ASSUMPTION 2.1. The limits A and [Omega] are finite. Also for every p we can find a bound [B.sub.p] such that the [p.sup.th] covariant derivatives of the curvature tensors [Rm.sub.k] of the metrics [G.sub.k] satisfy.

[absolute value of [D.sup.p][Rm.sub.k]] [less than or equal to] [B.sub.p],

the length being measured in [G.sub.k].

The previous authors have exerted great ingenuity to obtain the best possible regularity of the limit without the use of bounds on the derivatives of the curvature. Here all this cleverness is wasted. Thus the reader who only wishes to understand this case can spare himself all the machinery of harmonic coordinates and Sobolev estimates.

There are however other new problems to face. First off there is the injectivity radius. Clearly we need to keep some control over it, or the limit may collapse to a lower dimensional manifold or worse. Actually it would be very nice if we could let this happen and still get something in the limit which we could recognize as a solution to the Ricci flow. Perhaps some ambitious reader will wish to peruse the papers of Fukawa [F1] and [F2] with this in mind. Still the present result will be usable in some cases. In particular we want to be able to consider the limit of blow-ups of singularities such as a neck pinch. In this case, although each [M.sub.k] is compact, the limit M is a cylinder, which is not. Thus we studiously avoid any diameter bound. This raises the interesting question of what happens to the injectivity radius at a point P as the distance of P from the origin Q goes to infinity. Fortunately there is a very nice answer in the paper of Cheeger, Gromov and Taylor [GG&T], where they show that the injectivity radius [Rho](P) at P falls off at worst exponentially; in particular

[Rho](P) [greater than or equal to] c/[square root of B] [([Delta][square root of B]).sup.n] [e.sup.-C[square root of Bd(P,Q)]]

where B is an upper bound on the absolute value of the sectional curvatures, [Delta] a lower bound on the injectivity radius at Q with [Delta] [less than or equal to] c/[square root of B], and c [greater than] 0 and C [less than] [infinity] are constants depending only on the dimension while d(P, Q) is the distance from P to Q and n is the dimension. (Their statement and proof are quite elegant, involving a comparison with the volume of hyperbolic space. The form given here follows from knowing that the volume of the ball of radius r in hyperbolic space grows exponentially in r.) Considering a hyperbolic tube shrinking down at infinity shows that the exponential decay of the injectivity radius is possible. Fortunately such an exponential decay rate is not so fast as to seriously annoy any of the usual constructions involved in coveting a Riemannian manifold with suitable coordinate balls; it is only necessary to let the radii of the balls shrink exponentially as their centers go to infinity. This technique avoids having to assume a uniform lower bound for the injectivity radius over the whole manifold, a hypothesis which would be much harder to satisfy in applications.

The next step is to extract a subsequence of the metrics at t = 0 which will converge. This is a purely Riemannian problem and involves nothing from the Ricci flow, except that we allow ourselves the use of derivative bounds on the curvature. We state the result as a definition and a theorem.

Definition 2.2. A sequence [M.sub.k] = ([M.sub.k], [G.sub.k], [Q.sub.k], [q.sub.k]) of complete marked Riemannian manifolds converges to a complete marked Riemannian manifold M = (M, G, Q, q) if we can find a sequence of open sets [U.sub.k] in M containing Q and a sequence of diffeomorphisms [F.sub.k] of [U.sub.k] in M to open sets [V.sub.k] in [M.sub.k] taking Q to [Q.sub.k] and q to [q.sub.k], such that every compact set in M is eventually in all the [U.sub.k] and the pull-backs [Mathematical Expression Omitted] of the metrics [G.sub.k] by the maps [F.sub.k] converge on every compact subset of M uniformly together with all their derivatives.

THEOREM 2.3. Given any sequence [M.sub.k] = ([M.sub.k], [G.sub.k], [Q.sub.k], [q.sub.k]) as before such that we have uniform bounds on the Riemannian curvature tensors [RM.sub.k] and their [p.sup.th] covariant derivatives

[absolute value of [D.sup.p] [Rm.sub.k]] [less than or equal to] [B.sub.p]

for all p [greater than or equal to] 0 with constants [B.sub.p] independent of k, and such that we have uniform lower bounds on the injectivity radii [Rho]([Q.sub.k]) of the manifolds [M.sub.k] at the origins [Q.sub.k]

[Rho]([Q.sub.k]) [greater than or equal to] [Delta]

for some [Delta] [greater than] 0 independent of k, we can find a convergent subsequence.

We will prove this result in the next section. Here we show how to use it on solutions to the Ricci flow. Given a sequence [G.sub.k] of complete marked solutions to the Ricci flow on [M.sub.k] x (A, [Omega]), with curvature and injectivity radius bounds we can by the previous theorem find a subsequence which converges at t = 0 to a metric G on a manifold M. Then we get a sequence of open sets [U.sub.k] in M containing Q and a sequence of maps [F.sub.k] of [U.sub.k] in M to open sets [V.sub.k] in [M.sub.k] taking Q to [Q.sub.k] and q to [q.sub.k], such that the pull-backs [Mathematical Expression Omitted] of the metrics [G.sub.k] at time t = 0 by the maps [F.sub.k] converge to G uniformly together with all derivatives. However, the pull-backs [Mathematical Expression Omitted] are defined at all times A [less than] t [less than] [Omega] (although G is not, yet). We also have uniform bounds on the curvatures of the [Mathematical Expression Omitted] and their derivatives, independent of k. What we claim next is that we can find uniform bounds on all the covariant derivatives of the [Mathematical Expression Omitted] taken with respect to the fixed metric G.

LEMMA 2.4. Let M be a Riemannian manifold with metric G, K a compact subset of M, and [G.sub.k] a collection of solutions to the Ricci flow defined on neighborhoods of K x [[Beta], [Psi]] with the time interval [[Beta], [Psi]] containing t = 0. Let D denote the covariant derivative with respect to G and [Mathematical Expression Omitted] the length of a tensor with respect to G, while [D.sub.k] and [Mathematical Expression Omitted] are the same for [G.sub.k]. Suppose that

(a) the metrics [G.sub.k] are all uniformly equivalent to G at t = 0 on K, so that

cG(V, V) [less than or equal to] [G.sub.k](V, V) [less than or equal to] CG(V, V)

for some constants c [greater than] 0 and C [less than] [infinity] independent of k; and

(b) the covariant derivatives of the metrics [G.sub.k] with respect to the metric G are all uniformly bounded at t = 0 on K, so that

[absolute value of [D.sup.p][G.sub.k]] [less than or equal to] [C.sub.p]

for some constants [C.sub.p] [less than] [infinity] independent of k for all p [greater than or equal to] 1; and

(c) the covariant derivatives of the curvature tensors [Rm.sub.k] of the metrics [G.sub.k] are uniformly bounded with respect to the [G.sub.k] on K x [[Beta], [Psi]], so that

[Mathematical Expression Omitted]

for some constants [C[prime].sub.p] independent of k for all p [greater than or equal to] 0.

Then the metrics [G.sub.k] are uniformly bounded with respect to G on K x [[Beta], [Psi]], so that

[Mathematical Expression Omitted]

for some constants [Mathematical Expression Omitted] and [Mathematical Expression Omitted] independent of k, and the covariant derivatives of the metrics [G.sub.k] with respect to the metric G are uniformly bounded on K x [[Beta], [Psi]], so that

[Mathematical Expression Omitted]

for some constants [Mathematical Expression Omitted] independent of k with [Mathematical Expression Omitted], [Mathematical Expression Omitted] and [Mathematical Expression Omitted] depending only on c, C, [C.sub.p] and [C[prime].sub.p] and the dimension.

Proof. First we use the fact that the metrics [G.sub.k] are solutions to the Ricci flow to get bounds

[Mathematical Expression Omitted]

on K x [[Beta], [Psi]] for some constants [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. Since

[Delta]/[Delta]t [G.sub.k](V, V) = -2R[c.sub.k](V, V)

and we have

[absolute value of R[c.sub.k](V, V)] [less than or equal to] [C[prime].sub.0] [where] [G.sub.k] (V, V)

which gives

[absolute value of [Delta]/[Delta]t ln [G.sub.k](V, V)] [less than or equal to] 2 [C[prime].sub.0]

Now the bounds on ln [G.sub.k](V, V) at t = 0 easily extend to [Beta] [less than or equal to] t [less than or equal to][Psi].

Next we observe that the evolution of the connection is given by the first covariant derivative of the Ricci curvature, so that there is a formula of the form

[Delta]/[Delta]t [[Gamma].sub.k] = [DR[c.sub.k].

The connection [[Gamma].sub.k] of the metric [G.sub.k] is not a tensor, but if [Gamma] is the connection of G then the difference [[Gamma].sub.k] - [Gamma] is a tensor. Taking [Gamma] to be fixed in time, we get

[Delta]/[Delta]t ([[Gamma].sub.k] - [Gamma]) = DR[c.sub.k].

This leads to an estimate

[Mathematical Expression Omitted]

for a constant [C[double prime].sub.1] depending only on [C[prime].sub.1] and the dimension. But the tensor [[Gamma].sub.k] - [Gamma] is equivalent to the tensor D[G.sub.k], and this tensor is bounded at t = 0 by

[absolute value of D[G.sub.k]] [less than or equal to] [C.sub.1].

Integrating over time we get bounds

[Mathematical Expression Omitted]

for all of [Beta] [less than or equal to] t [less than or equal to] [Psi], using the fact that we already know [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are equivalent.

Next we want to bound [D.sup.2][G.sub.k]. Regarding D as fixed in time, we see that [Delta]/[Delta]t commutes with D. Thus

[Delta]/[Delta]t [D.sup.2][G.sub.k] = -2[D.sup.2][G.sub.k] = 2[D.sup.2]R[c.sub.k].

Now we can write

[Mathematical Expression Omitted].

We already have bounds on [Mathematical Expression Omitted] in [Mathematical Expression Omitted] and hence in [Mathematical Expression Omitted]. We also know that there is a formula

(D - [D.sub.k])R[c.sub.k] = D[G.sub.k] * R[c.sub.k]

where * denotes some inner tensor product, since [[Gamma].sub.k] - [Gamma] is equivalent to D[G.sub.k].

Then there is some formula

D(D - [D.sub.k])R[c.sub.k] = [D.sup.2][G.sub.k] * R[c.sub.k] + D[G.sub.k] * DR[c.sub.k].

We already have bounds on R[c.sub.k], DR[c.sub.k] and D[G.sub.k]. Finally there is a formula

(D - [D.sub.k])DR[c.sub.k] = D[G.sub.k] * DR[c.sub.k]

also. Combining we get an estimate

[absolute value of [Delta]/[Delta]t[D.sup.2][G.sub.k]] [less than or equal to] [C.sup.#] [absolute value of [D.sup.2][G.sub.k]] + [C.sup.#]

for some constant [C.sup.#]. Then our bounds on [absolute value of [D.sup.2][G.sub.k]] at t = 0 integrate to bounds on all of [Beta] [less than or equal to] t [less than or equal to] [Psi]. This gives

[Mathematical Expression Omitted]

for some constant [Mathematical Expression Omitted] independent of k. The bounds on the higher derivatives are the same. This proves the lemma.

We apply the lemma to the pull-back metrics [Mathematical Expression Omitted]. Since all of the [Mathematical Expression Omitted] are uniformly bounded with respect to the fixed metric G, we can find a subsequence which converges uniformly together with all its derivatives on every compact subset of M x (A, [Omega]). The limit metric will agree with that obtained previously at t = 0, where we knew it converged already. The limit G is now clearly itself a solution of the Ricci flow. This completes the proof of the Main Theorem, except for the proof of Theorem 2.3.

3. We turn now to the proof of Theorem 2.3. We are given a sequence [M.sub.k] = ([M.sub.k], [G.sub.k], [Q.sub.k], [q.sub.k]) of complete marked Riemannian manifolds with uniform bounds on the curvature and its covariant derivatives, and uniform lower bounds on the injectivity radii at the origins [Q.sub.k]. By our previous remark we can find a function [Rho](r) = [ce.sup.-Cr] such that if P is any point at distance r from the origin then the injectivity radius at P is large compared to [Rho](r); in fact let us make it bigger by a factor 500[[Gamma].sup.2] where [Gamma] = [e.sup.10cC].

In each [M.sub.k] we choose inductively a sequence of points [Mathematical Expression Omitted] for [Alpha] = 0, 1, 2, ... in the following way. First we let [Mathematical Expression Omitted]. The rest we choose so that if [Mathematical Expression Omitted] is the distance from [Mathematical Expression Omitted] and [Mathematical Expression Omitted], then [Mathematical Expression Omitted] is as small as possible, subject to the requirement that the open balls [Mathematical Expression Omitted] around [Mathematical Expression Omitted] of radius [Mathematical Expression Omitted] are all disjoint.

LEMMA 3.1. The balls [Mathematical Expression Omitted] cover [M.sub.k]. In fact for any r we can find [Lambda](r) independent of k such that the given balls for [Alpha] [less than or equal to] [Lambda](r) cover the ball B([Q.sub.k], r) of radius r around [Q.sub.k].

Proof. Let P [element of] [M.sub.k] and let r be the distance from P to the origin [Q.sub.k] and let [Rho] = [Rho](r). Consider those [Alpha] with [Mathematical Expression Omitted]. Then [Mathematical Expression Omitted]. Using the curvature bound and the injectivity radius bound, each ball [Mathematical Expression Omitted] has volume at least [Epsilon][[Rho].sup.n] where [Epsilon] [greater than] 0 is some constant and n is the dimension. These balls are all disjoint and contained in the ball B([Q.sub.k], r + c) since each [Mathematical Expression Omitted]. But we can estimate the volume of this ball from above by a function of r (exponential in fact). This gives us a bound [Lambda](r) on the number of [Alpha] with [Mathematical Expression Omitted]. Now I claim that the given point P must lie in one of the balls [Mathematical Expression Omitted]. If not, we could choose the next point in the sequence of [Mathematical Expression Omitted] to be P instead, for since [Mathematical Expression Omitted] the [Mathematical Expression Omitted] with [r.sub.k] [less than or equal to] r. But this is a contradiction. This proves the lemma.

Now it is desirable from the beginning to work with balls of a standard size. To this end observe that [Mathematical Expression Omitted] because with all [Mathematical Expression Omitted] this is as far out as we would have to go to find [Mathematical Expression Omitted]. Hence by passing to a subsequence using a diagonalization argument we may assume that [Mathematical Expression Omitted] converges to some [r.sup.[Alpha]] for each [Alpha]. Then [Mathematical Expression Omitted] converges to [[Rho].sup.[Alpha]] = [Rho]([r.sup.[Alpha]]). Then for all [Alpha] we can find k([Alpha]) such that if k [greater than or equal to] k([Alpha]) then [Mathematical Expression Omitted] and [Mathematical Expression Omitted].

LEMMA 3.2. For every r there exists a number k(r) such that if k [greater than or equal to] k(r) then the balls [Mathematical Expression Omitted] for [Alpha] [less than or equal to] [Lambda](r) cover the ball B([Q.sub.k], r) of radius r around the origin.

Proof. Let k(r) = max{k([Alpha]): [Alpha] [less than or equal to] [Lambda](r)}. Then [Mathematical Expression Omitted] and

[Mathematical Expression Omitted]

so the balls [Mathematical Expression Omitted] for [Alpha] [less than or equal to] [Lambda](r) cover B([Q.sub.k], r) by Lemma 3.1.

Suppose that [Mathematical Expression Omitted] and [Mathematical Expression Omitted] meet for k [greater than or equal to] k([Alpha]) and k [greater than or equal to] k([Beta]). Then we must have

[Mathematical Expression Omitted]

and since [[Rho].sup.[Alpha]] [less than or equal to] c and [[Rho].sup.[Beta]] [less than or equal to] c we get

[Mathematical Expression Omitted].

This then makes

[Mathematical Expression Omitted]

with [Gamma] = [e.sup.8Cc], since [Rho](r) = [ce.sup.-Cr]. It follows that

[[Rho].sup.[Beta]] [less than or equal to] 4 [Gamma][[Rho].sup.[Alpha]].

Hence if we let [Mathematical Expression Omitted] then [Mathematical Expression Omitted] whenever [Mathematical Expression Omitted] and [Mathematical Expression Omitted] meet and k [greater than or equal to] k([Alpha]) and k [greater than or equal to] k([Beta]).

Next we define balls [Mathematical Expression Omitted] and [Mathematical Expression Omitted] and [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. Since we have

[Mathematical Expression Omitted]

the [Mathematical Expression Omitted] disjoint. Since [Mathematical Expression Omitted] the [Mathematical Expression Omitted] cover B([Q.sub.k], r) for [Alpha] [less than or equal to] [Lambda](r) as before. If [Mathematical Expression Omitted] meets [Mathematical Expression Omitted] with k, l [greater than or equal to] k([Alpha]) then

[Mathematical Expression Omitted]

and since [[Rho].sup.[Alpha]] [[Rho].sup.[Beta]] [less than or equal to] c we get

[Mathematical Expression Omitted]

and this makes

[Mathematical Expression Omitted]

since [Mathematical Expression Omitted] and [Rho](r) = [c.sup.-Cr. This makes

[[Rho].sup.[Beta]] [less than or equal to] 4[Gamma][[Rho].sup.[Alpha]].

Now any point in [Mathematical Expression Omitted] has distance at most

5[[Rho].sup.[Alpha]] + 5[[Rho].sup.[Beta]] + 5[[Rho].sup.[Beta]] [less than or equal to] 45[Gamma][[Rho].sup.[Alpha]]

for [Mathematical Expression Omitted] and hence [Mathematical Expression Omitted]. Likewise any point in [Mathematical Expression Omitted] has distance at most

5[[Rho].sup.[Alpha]] + 5[[Rho].sup.[Beta]] + 45[Gamma][[Rho].sup.[Beta]] [less than or equal to] 205[[Gamma].sup.2][[Rho].sup.[Alpha]]

and hence [Mathematical Expression Omitted]. Since the injectivity radius at [Mathematical Expression Omitted] is larger than [Mathematical Expression Omitted] and [Mathematical Expression Omitted] we see that [Mathematical Expression Omitted] is still a nice embedded ball.

LEMMA 3.3. There exists a number N such that for any given [Alpha] there are at most N choices of [Beta] for which [Mathematical Expression Omitted] meets [Mathematical Expression Omitted] when k [greater than or equal to] max{k([Alpha]), k([Beta])}.

Proof. If [Mathematical Expression Omitted] meets [Mathematical Expression Omitted] then [Mathematical Expression Omitted] and the balls [Mathematical Expression Omitted] are all disjoint. Moreover [[Rho].sup.[Beta]] is at least a fraction of [[Rho].sup.[Alpha]] as before. Thus (using the curvature bound and using [[Rho].sup.[Alpha]] [less than or equal to] c) each [Mathematical Expression Omitted] has volume at least a fraction of [([[Rho].sup.[Alpha]]).sup.n] where n is the dimension, while [Mathematical Expression Omitted] has volume at most a multiple of [([[Rho].sup.[Alpha]]).sup.n]. The ratio of these numbers gives a bound N. This proves the lemma.

Next observe that by passing to a subsequence we can guarantee that for any pair [Alpha] and [Beta] we can find a number k([Alpha], [Beta]) such that if k [greater than or equal to] k([Alpha], [Beta]) then either [Mathematical Expression Omitted] always meets [Mathematical Expression Omitted] or it never does. Let us increase k(r) if necessary so that k(r) [greater than or equal to] k([Alpha], [Beta]) if [Alpha], [Beta] [less than or equal to] [Lambda](r). Then we have achieved the following situation.

LEMMA 3.4. For every r we can find [Lambda](r) and k(r) such that if k [greater than or equal to] k(r) then the ball B([Q.sub.k], r) of radius r around the origin in [M.sub.k] is covered by the balls [Mathematical Expression Omitted] for [Alpha] [less than or equal to] [Lambda](r). Moreover no more than N balls ever meet, and for any [Alpha], [Beta] [less than or equal to][Lambda](r) either [Mathematical Expression Omitted] meets [Mathematical Expression Omitted] for all k [greater than or equal to] k(r) or for none.

Now we let [Mathematical Expression Omitted], [E.sup.Alpha], [Mathematical Expression Omitted] and [Mathematical Expression Omitted] be the balls of radii 4[[Rho].sup.[Alpha]], 5[[Rho].sup.[Alpha]], 45[Gamma][[Rho].sup.[Alpha]], and 205[[Gamma].sup.2][[Rho].sup.[Alpha]] around the origin in Euclidean space. At each point [Mathematical Expression Omitted] we choose an orthonormal frame [Mathematical Expression Omitted], taking care to pick [Mathematical Expression Omitted] at [Mathematical Expression Omitted], but choosing the others as we please. Then we define coordinate charts [Mathematical Expression Omitted] as the composition of the linear isometry of Euclidean space to the tangent space at [Mathematical Expression Omitted] taking the standard frame to the frame [Mathematical Expression Omitted] with the exponential map at [Mathematical Expression Omitted] into [M.sub.k]. We also get maps [Mathematical Expression Omitted] and [Mathematical Expression Omitted] in the same way. We let [Mathematical Expression Omitted] (and [Mathematical Expression Omitted] and [Mathematical Expression Omitted]) be the pull-backs of the metrics [G.sub.k] on [M.sub.k] by [Mathematical Expression Omitted]. We also consider the coordinate transition functions [Mathematical Expression Omitted] and [Mathematical Expression Omitted] defined by

[Mathematical Expression Omitted].

Clearly [Mathematical Expression Omitted]. Moreover [Mathematical Expression Omitted] is an isometry from [Mathematical Expression Omitted] to [Mathematical Expression Omitted] and [Mathematical Expression Omitted] from [Mathematical Expression Omitted] to [Mathematical Expression Omitted]. Since the metrics [Mathematical Expression Omitted] are in geodesic coordinates and have their curvatures and their covariant derivatives uniformly bounded, it follows by Corollary 4.10 in the next section that by passing to a subsequence we can guarantee that for each [Alpha] (and indeed for all [Alpha] by diagonalization) the metrics [Mathematical Expression Omitted] converge uniformly with all their derivatives to a smooth metric [G.sup.[Alpha]] on [E.sup.[Alpha]] (or [Mathematical Expression Omitted] or [Mathematical Expression Omitted]) which is also in geodesic coordinates.

Look now at any pair [Alpha], [Beta] for which the balls [Mathematical Expression Omitted] and [Mathematical Expression Omitted] always meet for large k, and thus the maps [Mathematical Expression Omitted] (and [Mathematical Expression Omitted] and [Mathematical Expression Omitted] and [Mathematical Expression Omitted]) are always defined for large k. By Theorem 5.1 it follows that the isometries [Mathematical Expression Omitted] (and [Mathematical Expression Omitted] and [Mathematical Expression Omitted] and [Mathematical Expression Omitted]) always have a convergent subsequence, so we may assume [Mathematical Expression Omitted] (and [Mathematical Expression Omitted] and [Mathematical Expression Omitted] and [Mathematical Expression Omitted]). The limit maps [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are isometries in the limit metrics [G.sup.[Beta]] and [G.sup.[Alpha]]. Moreover

[Mathematical Expression Omitted].

We are now done picking subsequences, and everything is prepared to converge.

Definition 3.5. We say that a diffeomorphism F between Riemannian manifolds is an ([[Epsilon].sub.1], [[Epsilon].sub.2],..., [[Epsilon].sub.p]) approximate isometry if

[absolute value of tDF [center dot] DF - I] [less than] [[Epsilon].sub.1]

and

[absolute value of [D.sup.2]F] [less than] [[Epsilon].sub.2],..., [absolute value of [D.sup.p]F] [less than] [[Epsilon].sub.p]

where [D.sup.p]F is the [p.sup.th] covariant derivative of F with respect to the given metrics (regarded as a p-multilinear map of the tangent bundle of the domain into the pull-back by F of the tangent bundle of the range).

THEOREM 3.6. Take the subsequence [M.sub.k] = ([M.sub.k], [G.sub.k], [Q.sub.k], [q.sub.k]) chosen above. Then for every r and every ([[Epsilon].sub.1], [[Epsilon].sub.2],..., [[Epsilon].sub.p]), and for all k and l sufficiently large in comparison, we can find a diffeomorphism [F.sub.kl] of a neighborhood of the ball [B.sub.k](r) of radius r around [Q.sub.k] in [M.sub.k] into an open set in [M.sub.l] which is an ([[Epsilon].sub.1], [[Epsilon].sub.2],...,[[Epsilon].sub.p]) approximate isometry.

Proof. The idea (following Peters [P1] or Greene and Wu [G& W]) is to define the map [Mathematical Expression Omitted] of [Mathematical Expression Omitted] to [Mathematical Expression Omitted] for k and l large compared to [Alpha] so as to be the identity map on [E.sup.[Alpha]] in the coordinate charts [Mathematical Expression Omitted] and [Mathematical Expression Omitted], and then to define [F.sub.kl] on a neighborhood of B([Q.sub.k], r) for k, l [greater than or equal to] k(r) by averaging the maps [Mathematical Expression Omitted] for [Alpha] [less than or equal to] [Lambda](r). To describe the averaging process on [Mathematical Expression Omitted] we only need to consider those [Mathematical Expression Omitted] with [Beta] [less than or equal to] [Lambda](r) which meet [Mathematical Expression Omitted]; there are never more than N of them, and they are the same for k and l when k, l [greater than or equal to] k(r). The averaging process is defined by taking [F.sub.kl](X) to be the center of mass of the [Mathematical Expression Omitted] for [Mathematical Expression Omitted] averaging over those [Beta] where [Mathematical Expression Omitted] meets [Mathematical Expression Omitted] using weights [Mathematical Expression Omitted] defined by a partition of unity. The center of mass of the points [Y.sup.[Beta]] with weights [[Mu].sup.[Beta]] is defined to be the point Y such that

[exp.sub.Y] [V.sup.[Beta]] = [Y.sup.[Beta]] and [summation of] [[Mu].sup.[Beta]][V.sup.[Beta]] = 0.

When the points [Y.sup.[Beta]] are all close and the weights [[Mu].sup.[Beta]] satisfy 0 [less than or equal to] [[Mu].sup.[Beta]] [less than or equal to] 1 then there will be a unique solution Y close to [Y.sup.[Beta]] which depends smoothly on the [Y.sup.[Beta]] and the [[Mu].sup.[Beta]] (see [G&K] for the details). The point Y is found by the inverse function theorem, which also provides bounds on all the derivatives of Y as a function of the [Y.sup.[Beta]] and the [[Mu].sup.[Beta]].

Since [Mathematical Expression Omitted] and [Mathematical Expression Omitted], the map [Mathematical Expression Omitted] can be represented in local coordinates by the map

[Mathematical Expression Omitted]

defined by

[Mathematical Expression Omitted].

Since [Mathematical Expression Omitted] as k [approaches] [infinity] and [Mathematical Expression Omitted] as l [approaches] [infinity] and [Mathematical Expression Omitted], we see that the maps [Mathematical Expression Omitted] as k, l [approaches] [infinity] for each choice of [Alpha] and [Beta]. The weights [Mathematical Expression Omitted] are defined in the following way. We pick for each [Beta] a smooth function [[Psi].sup.[Beta]] which equals 1 on [Mathematical Expression Omitted] and equals 0 outside [E.sup.[Beta]]. We then transfer [[Psi].sup.[Beta]] to a function [Mathematical Expression Omitted] on [M.sub.k] by the coordinate map [Mathematical Expression Omitted]. Then let

[Mathematical Expression Omitted]

as usual. In the coordinate chart [E.sup.[Alpha]] the function [Mathematical Expression Omitted] looks like the composition of [Mathematical Expression Omitted] with [[Psi].sup.[Beta]]. Call this function

[Mathematical Expression Omitted].

Then as k [approaches] [infinity], [Mathematical Expression Omitted] where

[[Psi].sup.[Alpha][Beta]] = [[Psi].sup.[Beta]] [convolution] [J.sup.[Beta][Alpha]].

In the coordinate chart [E.sup.[Alpha]] the function [Mathematical Expression Omitted] looks like

[Mathematical Expression Omitted]

and [Mathematical Expression Omitted] as k [approaches] [infinity] where

[[Mu].sup.[Alpha][Beta]] = [[Psi].sup.[Alpha][Beta]] / [summation over [Gamma]] [[Psi].sup.[Alpha][Gamma]].

Since the sets [Mathematical Expression Omitted] cover B([Q.sub.k], r), it follows that [Mathematical Expression Omitted] on this set and hence [Mathematical Expression Omitted] there is no problem bounding all these functions and their derivatives. There is a small problem in that we want to guarantee that the averaged map still takes [Q.sub.k] to [Q.sub.l] and [q.sub.k] to [q.sub.l]. This is true at least for the map [Mathematical Expression Omitted]. Therefore it will suffice to guarantee that [Mathematical Expression Omitted] in a neighborhood of [Q.sub.k] if [Alpha] [not equal to] 0. This happens if the same is true for [Mathematical Expression Omitted]. If not, we can always replace [Mathematical Expression Omitted] by [Mathematical Expression Omitted] which still leaves [Mathematical Expression Omitted] or [Mathematical Expression Omitted] everywhere, and this is sufficient to make [Mathematical Expression Omitted] everywhere. Now in the local coordinates [E.sup.[Alpha]] we are averaging maps [Mathematical Expression Omitted] which converge to the identity with respect to weights [Mathematical Expression Omitted] which converge. It follows that the averaged map converges to the identity in these coordinates. Thus [F.sub.kl] can be made to be an ([[Epsilon].sub.1], [[Epsilon].sub.2], ..., [[Epsilon].sub.p]) approximate isometry on B([Q.sub.k], r) when k and l are suitably large. At least the estimates

[absolute value of tD[F.sub.kl] [multiplied by] D[F.sub.kl] - I] [less than] [[Epsilon].sub.1]

and [absolute value of [D.sup.2][F.sub.kl]] [less than] [[Epsilon].sub.2], ..., [absolute value of [D.sup.p][F.sub.kl]] [less than] [[Epsilon].sub.p] on B([Q.sub.k], r) follow from the local coordinates. We still need to check that [F.sub.kl] is a diffeomorphism on a neighborhood of B([Q.sub.k], r).

This, however, follows quickly enough from the fact that we also get a map [F.sub.lk] on a slightly larger ball B([Q.sub.l], r[prime]) which contains the image of [F.sub.lk] on B([Q.sub.k], r) if we take r[prime] = (1 + [[Epsilon].sub.1])r, and [F.sub.lk] also satisfies the above estimates. Also [F.sub.kl] and [F.sub.lk] fix the markings, so the composition [F.sub.lk] [convolution] [F.sub.kl] satisfies the same sort of estimates and fixes the origin [Q.sub.k] and the frame [q.sub.k]. Then [F.sub.lk] [convolution] [F.sub.kl] must be very close to the identity on B([Q.sub.k], r), and it follows that [F.sub.kl] is invertible. For if [F.sub.kl](X) = [F.sub.kl](Y), then we conclude that X is very close to Y. But on each ball [E.sub.[Alpha]] in local coordinates [F.sub.kl] is almost the identity, so once X and Y fall in the same ball [Mathematical Expression Omitted] they must agree. Now X lies in some [Mathematical Expression Omitted], and if the distance from X to Y is no more than [[Rho].sup.[Alpha]] then X and Y both lie in [Mathematical Expression Omitted]. If we need [Mathematical Expression Omitted] to help cover B([Q.sub.k], r) then [Mathematical Expression Omitted] and since [[Rho].sup.[Alpha]] [less than or equal to] c we get [Mathematical Expression Omitted] and [Mathematical Expression Omitted] with [Rho] = [Rho](r). Then [Mathematical Expression Omitted], and this provides a lower bound on [[Rho].sup.[Alpha]] in terms of r only. So choosing ([[Epsilon].sub.1], [[Epsilon].sub.2], ..., [[Epsilon].sub.p]) small compared to r will make X and Y close enough to lie in some [Mathematical Expression Omitted] for [Alpha] [less than or equal to] [Lambda](r). This completes the proof of Theorem 3.2.

We now know the manifolds [M.sub.k] are nearly isometric for large k. We still need to construct the limit manifold. In the case where the diameters are uniformly bounded this is not a problem, since there are only a finite number of diffeomorphism types. In our case, however, life is not so easy. One can imagine a limit of surfaces with [M.sub.k] having k holes as k [right arrow] [infinity] to see the problem.

To handle this, we first note another problem, which is that the approximate isometries we constructed don't compose to give each other, that is

[F.sub.lm] [convolution] [F.sub.kl] [not equal] [F.sub.km].

We can, however, rectify this situation. For each integer r we choose the numbers ([[Epsilon].sub.1](r), [[Epsilon].sub.2](r), ..., [[Epsilon].sub.r](r)) so small that when we choose k(r) large in comparison and find the maps [F.sub.k(r),k(r + 1)] constructed above on neighborhoods of B([Q.sub.k(r)], r) in [M.sub.k(r)] into [M.sub.k(r + 1)] the image always lies in B([Q.sub.k(r + 1)], r + 1) and the composition of [F.sub.k(r),k(r + 1)] with [F.sub.k(r+1),k(r+2)] and ... and [F.sub.k(s - 1),k(s)] for any s [greater than] r is still an ([[Eta].sub.1](r), [[Eta].sub.2](r), ..., [[Eta].sub.r](r)) isometry for any choice of [[Eta].sub.j](r), say [[Eta].sub.j](r) = 1/r for 1 [less than or equal to] j [less than or equal to] r. Basically this is just the idea that by making the terms [A.sub.s] in a series small enough for s [greater than or equal to] r we can always make [summation over] s [greater than or equal to] r [A.sub.s] as small as we like. Now we simplify the notation by writing [M.sub.r] in place of [M.sub.k](r) and [F.sub.r] in place of [F.sub.k(r),k(r + 1)].

Then [F.sub.r] is a diffeomorphism

[F.sub.r] : B([Q.sub.r], r) [right arrow] B([Q.sub.r + 1], r + 1)

and the composition

[F.sub.s - 1] [convolution] ... [convolution] [F.sub.r] : B([Q.sub.r], r) [right arrow] B([Q.sub.s], s)

is always an ([[Eta].sub.1], ..., [[Eta].sub.r]) approximate isometry.

We now construct the limit manifold M as a topological space by identifying the balls B([Q.sub.r], r) with each other using the homeomorphisms [F.sub.r]. Given any two points X and Y in M, we have X [element of] B([Q.sub.r], r) and Y [element of] B([Q.sub.s], s) for some r and s. If r [less than or equal to] s then X [element of] B([Q.sub.s], s) also, by identification. A set in M is open if and only if it intersects each B([Q.sub.r], r) in an open set. Then choosing disjoint neighborhoods of X and Y in B([Q.sub.s], s) gives disjoint neighborhoods of X and Y in M. Thus M is a Hausdorff space.

Any smooth chart on B([Q.sub.r], r) also gives a smooth chart on B([Q.sub.s], s) for all s [greater than or equal to] r. The union of all such charts gives a smooth atlas on M. It is fairly easy to see that the metrics [G.sub.r] on B([Q.sub.r], r) converge to a smooth metric G on M uniformly together with all derivatives on compact sets. For since the [F.sub.r] are very good approximate isometries, the [G.sub.r] are very close to each other, and hence form a Cauchy sequence (together with their derivatives, in the sense that the covariant derivatives of [G.sub.r] with respect to [G.sub.s] are very small when r and s are both large). One checks in the usual way that such a Cauchy sequence converges. One can also see that the limit metric is complete. For if we look at the closed ball of radius r - 1 in [M.sub.r] around the origin [Q.sub.r], which is identified with the origin Q in the limit M, we see this ball is compact and contains the closed ball of radius r - 2 around Q in the limit M when r is large. But a closed subset of a compact set is compact, so all the closed balls around Q in M are compact. Thus M is complete.

The origins [Q.sub.r] are identified with each other, and hence with an origin Q in M. Likewise the frames [q.sub.r] in [M.sub.r] are identified with each other, and hence with a frame q in M. Now it is the inverses of the maps identifying B([Q.sub.r], r) with open subsets of M that provide the diffeomorphisms of open sets in M to the manifolds [M.sub.r] such that the pull-backs of the metrics [G.sub.r] converge to G. This completes the proof of Theorem 2.3, except we still owe the reader proof of Corollary 4.10.

4. In this section we give proofs of Theorems 4.1 and 4.2 quoted earlier. First we look at geodesic coordinates. Recall that a metric [g.sub.ij](x)d[x.sup.i]d[x.sup.j] defined in a ball around the origin is in geodesic coordinates if every line through the origin is a geodesic (parametrized proportional to arc length) and if [g.sub.ij] = [I.sub.ij] at the origin x = 0.

LEMMA 4.1. The metric [g.sub.ij] is in geodesic coordinates if and only if

[g.sub.ij][x.sup.i] = [I.sub.ij][x.sup.i].

Geometrically this says that the metric inner product with the radial vector agrees with the Euclidean inner product.

Proof. A curve [x.sup.i] = [x.sup.i](t) is a geodesic (parametrized proportional to arc length) if it satisfies the equation

[Mathematical Expression Omitted].

Then the line [x.sup.i] = t[v.sup.i] for fixed [v.sub.i] are geodesics if and only if

[Mathematical Expression Omitted]

which is one set of equations for geodesic coordinates. Suppose this is true. Since the Christoffel symbols are given by

[Mathematical Expression Omitted]

we see that

[x.sup.j][x.sup.k] [Delta]/[Delta][x.sup.l] [g.sub.jk] = 2[x.sup.j][x.sup.k] [Delta]/[Delta][x.sup.j] [g.sub.kl].

From this we compute [x.sup.i][x.sup.j][x.sup.k] [Delta]/[Delta][x.sup.i] [g.sub.jk] = 0 and

[x.sup.i] [Delta]/[Delta][x.sup.i] [[g.sub.jk][x.sup.j][x.sup.k]] = 2[g.sub.jk][x.sup.j][x.sup.k]

which can also be written as

[x.sup.i] [Delta]/[Delta][x.sup.i] ln [[g.sub.jk][x.sup.j][x.sup.k]] = 2.

But we also have

[x.sup.i] [Delta]/[Delta][x.sup.i] ln[[I.sub.jk][x.sup.j][x.sup.k]] = 2

(since [I.sub.jk] is also in geodesic coordinates). Therefore the ratio

[g.sub.jk][x.sup.j][x.sup.k]/[I.sub.jk][x.sup.j][x.sup.k]

is constant along radial lines. But [g.sub.jk] [right arrow] [I.sub.jk] as x [right arrow] 0 along any radial line, so the ratio must be identically one. Thus we have

[g.sub.jk][x.sup.j][x.sup.k] = [I.sub.jk][x.sup.j][x.sup.k].

If we differentiate this identity we get

[x.sup.j][x.sup.k] [Delta]/[Delta][x.sup.l] [g.sub.jk] + 2[g.sub.kl][x.sup.k] = 2[I.sup.kl][x.sup.k]

which by our previous equation gives

[x.sup.j][x.sup.k] [Delta]/[Delta][x.sup.j] [g.sub.kl] + [g.sub.kl][x.sup.k] = [I.sub.kl][x.sup.k].

But this implies that

[x.sup.j] [Delta]/[Delta][x.sup.j] [[g.sub.kl][x.sup.k] - [I.sub.kl][x.sup.k]] = 0

and hence the function [g.sub.kl][x.sup.k] - [I.sub.kl][x.sup.k] is constant along radial lines. But [g.sub.kl] [right arrow] [I.sub.kl] as x [right arrow] 0, so the function must vanish everywhere. Hence [G.sub.kl][x.sup.k] = [I.sub.kl][x.sup.k] in geodesic coordinates.

Conversely if [g.sub.kl][x.sup.k] = [I.sub.kl][x.sup.k], then differentiating we get

[x.sup.k] [Delta]/[Delta][x.sup.j] [g.sub.kl] + [g.sub.jl] = [I.sub.jl]

from which we conclude

[x.sup.k][x.sup.l] [Delta]/[Delta][x.sup.j][g.sub.kl] = 0 and [x.sup.j][x.sup.l] [Delta]/[Delta][x.sup.j] [g.sub.kl] = 0.

Then we get [Mathematical Expression Omitted] which shows we are in geodesic coordinates. This proves the lemma.

Note in particular that in geodesic coordinates

[[absolute value of x].sup.2] = [g.sub.ij][x.sup.i][x.sup.j] = [I.sub.ij][x.sup.k][x.sub.j]

is unambiguously defined. Also if we let [x.sup.i] = t[v.sup.i] then for t [not equal to] 0

[Mathematical Expression Omitted]

and then letting t [right arrow] 0 we conclude that [Mathematical Expression Omitted]. Hence from

[Mathematical Expression Omitted]

we see that all the first derivatives of [g.sub.jk] vanish at the origin.

Introduce the symmetric tensor

[A.sub.ij] = 1/2 [x.sup.k] [Delta]/[Delta][x.sup.k] [g.sub.ij].

Since we have [g.sub.jk][x.sup.k] = [I.sub.jk][x.sup.k], we get

[x.sup.k] [Delta]/[Delta][x.sup.i] [g.sub.jk] = [I.sub.ij] - [g.sub.ij] = [x.sup.k] [Delta]/[Delta][x.sup.j] [g.sub.ik]

and hence from the formula for [Mathematical Expression Omitted]

[Mathematical Expression Omitted].

Hence [A.sub.kl][x.sup.k] = 0. Let [D.sub.i] be the covariant derivative with respect to the metric [g.sub.ij]. Then

[Mathematical Expression Omitted].

Introduce the potential function

p = [[absolute value of x].sup.2]/2 = 1/2[g.sub.ij][x.sup.i][x.sup.j].

We can use the formulas above to compute

[D.sub.i]p = [g.sub.ij][x.sup.j].

Also we get

[D.sub.i][D.sub.j]p = [g.sub.ij] + [A.sub.ij].

The defining equation for p gives

[g.sup.ij][D.sub.i]p[D.sub.j]p = 2p.

If we take the covariant derivative of this equation we get

[g.sup.kl][D.sub.j][D.sub.k]p [convolution] [D.sub.l]p = [D.sub.j]p

which is equivalent to [A.sub.jk][x.sup.k] = 0. But if we take the covariant derivative again we get

[g.sup.kl][D.sub.i][D.sub.j][D.sub.k]p [convolution] [D.sub.l]p + [g.sup.kl][D.sub.j][D.sub.k]p [convolution] [D.sub.i][D.sub.l]p = [D.sub.i][D.sub.j]p.

Now switching derivatives

[D.sub.i][D.sub.j][D.sub.k]p = [D.sub.i][D.sub.k][D.sub.j]p = [D.sub.k][D.sub.i][D.sub.j]p + [R.sub.ikjl][g.sup.lm][D.sub.m]p

and if we use this and [D.sub.i][D.sub.j]p = [g.sub.ij] + [A.sub.ij] and [g.sup.kl][D.sub.l]p = [x.sup.k] we find that

[x.sup.k][D.sub.k][A.sub.ij] + [A.sub.ij] + [g.sup.kl][A.sub.ik][A.sub.jl] + [R.sub.ikjl][x.sup.k][x.sup.l] = 0.

Let us now use [absolute value of [T.sub.i...j]] to denote the length of a tensor with respect to the metric [g.sub.ij], so that

[[absolute value of [T.sub.i...j]].sup.2] = [g.sup.ik] ... [g.sup.jl][T.sub.i...j][T.sub.k...l].

From our assumed curvature bounds we can take [absolute value of [R.sub.ijkl]] [less than or equal to] [B.sub.0]. Then we get the following estimate.

LEMMA 4.1. We have

[absolute value of [x.sup.k][D.sub.k][A.sub.ij] + [A.sub.ij]] [less than or equal to] C[[absolute value of [A.sub.ij]].sup.2] + C[B.sub.0][r.sup.2]

on the ball [absolute value of x] [less than or equal to] r for some constant C depending only on the dimension.

We now show how to use the maximum principle on such equations.

THEOREM 4.2. Let f be a function on a ball [absolute value of x] [less than or equal to] r, and let [Lambda] [greater than] 0 be a constant. Then

[lambda] sup [absolute value of f] [less than or equal to] sup [absolute value of [x.sup.k] [Delta]f/[Delta][x.sup.k] + [Lambda]f].

Proof. We can assume the sup is positive by interchanging f with -f if necessary. If it occurs in the interior, we can prove the estimate on a smaller ball. If it occurs on the boundary, then since [x.sup.k][Delta]/[Delta][x.sup.k] is pointing outward we have

[x.sup.k] [Delta]f/[Delta][x.sup.k] [greater than or equal to] 0

at that point. The result follows.

COROLLARY 4.3. For any tensor T = {[T.sub.i...j]} and any constant [Lambda] [greater than] 0 we have

[Lambda] sup [absolute value of T] [less than or equal to] sup [absolute value of [x.sup.k][D.sub.k]T = [Lambda]T].

Proof. Apply the preceding theorem to f = [[absolute value of T].sup.2] and get

[Mathematical Expression Omitted]

and expand the latter as twice the inner product of T with [x.sup.k][D.sub.k]T + [Lambda]T. This gives

2[Lambda] sup [[absolute value of T].sup.2] [less than or equal to] 2 sup [absolute value of T] sup [absolute value of [x.sup.k][D.sub.k]T + T]

and we can divide by 2 sup [absolute value of T] to get the result.

We apply this to the tensor [A.sub.ij].

COROLLARY 4.4. On the ball [absolute value of x] [less than or equal to] r we get

[Mathematical Expression Omitted]

for some constant depending only on the dimension.

THEOREM 4.5. There exist constants c [greater than] 0 and [C.sub.0] [less than] [infinity] such that if the metric [g.sub.ij] is in geodesic coordinates with [absolute value of [R.sub.ijkl]] [less than or equal to] [B.sub.0] in the ball of radius r [less than or equal to]c/[square root of [B.sub.0]] then

[absolute value of [A.sub.ij]] [less than or equal to] [C.sub.0][B.sub.0][r.sup.2].

Proof. We use a barrier argument. Since the derivatives of [g.sub.ij] vanish at the origin, so does [A.sub.ij]. Hence the estimate holds near the origin. But Corollary 4.4 says that [absolute value of [A.sub.ij]] avoids an interval when c is chosen small. In fact the inequality

X [less than or equal to] C[X.sup.x] + D

is equivalent to

[absolute value of 2CX - 1] [greater than or equal to] [square root of 1 - 4CD]

which makes X avoid an interval if 4CD [less than] 1. (Hence in our case we need to choose c with 4[C.sup.2][c.sup.2] [less than] 1.) Then if X is on the side containing 0 we get

X [less than or equal to] 1 - [square root of 1 - 4CD]/2c [less than or equal to] 2D.

This gives [absolute value of [A.sub.ij]] [less than or equal to] [C.sub.0][B.sub.0][r.sup.2] with [C.sub.0] = 2C.

We can also derive bounds on all the covariant derivatives of p in terms of bounds on the covariant derivatives of the curvature. To simplify the notation, we let

[D.sup.q]p = {[D.sub.j1][D.sub.j2] ... [D.sub.jq]p}

denote the [q.sub.th] covariant derivative, and in estimating [D.sup.q]p we will lump all the lower order terms into a general slush term [[Phi].sup.q] which will be a polynomial in [D.sup.1]p, [D.sup.2]p, ..., [D.sup.q - 1]p and Rm, [D.sup.j]Rm, ..., [D.sup.q - 2]Rm. We already have estimates on a ball of radius r

p [less than or equal to] [r.sup.2]/2

[absolute value of [D.sup.1]p] [less than or equal to] r

[absolute value of [A.sub.ij]] [less than or equal to] [C.sub.0][B.sub.0][r.sup.2]

and since [D.sub.i][D.sub.j]p = [g.sub.ij] + [A.sub.ij] and r [less than or equal to] c / [square root of B] if we choose c small we can make

[absolute value of [A.sub.ij] [less than or equal to] 1/2.

and we get

[absolute value of [D.sup.2]p] [less than or equal to] [C.sub.2]

for some constant [C.sub.2] depending only on the dimension.

Start with the equation [g.sup.ij][D.sub.i]p[D.sub.j]p = 2p and apply repeated covariant derivatives. Observe that we get an equation which starts out

[g.sup.ij][D.sub.i]p[D.sup.q][D.sub.j]p + ... = 0

where the omitted terms only contain derivatives [D.sup.q]p and lower. If we switch two derivatives in a term [D.sup.q + 1]p or lower, we get a term which is a product of a covariant derivative of Rm of order at most q - 2 (since the two closest to p commute) and a covariant derivative of p of order at most q - 1; such a term can be lumped in with the slush term [[Phi].sup.q]. Therefore up to terms in [[Phi].sup.q] we can regard the derivatives as commuting. Then paying attention to the derivatives in [D.sup.q]p we get an equation

[g.sub.ij][D.sub.i]p[D.sub.j][D.sub.[k.sub.1]] ... [D.sub.[k.sub.q]]p + [g.sup.ij][D.sub.i][D.sub.[k.sub.1]]p[D.sub.j][D.sub.[k.sub.2]] ... [D.sub.[k.sub.q]]p

+ [g.sup.ij][D.sub.i][D.sub.[k.sub.2]]p[D.sub.j][D.sub.[k.sub.1]][d.sub.[k.sub.3]] ... [D.sub.[k.sub.q]]p

+ ...

+ [g.sup.ij][D.sub.i][D.sub.kq]p[D.sub.j][D.sub.[k.sub.1]] ... [D.sub.[k.sub.q - 1]p

= [D.sub.[k.sub.1]] ... [D.sub.[k.sub.q]]p + [[Phi].sup.q].

Recalling that [D.sub.i][D.sub.j]p = [g.sub.ij] + [A.sub.ij] we can rewrite this as

[g.sup.ij][D.sub.i]p[D.sub.j][D.sub.[k.sub.1]] ... [D.sub.[k.sub.q]]p + (q - 1)[D.sub.[k.sub.1]] ... [D.sub.[k.sub.q]]p

+ [g.sup.ij][A.sub.i[k.sub.1]][D.sub.j][D.sub.[k.sub.2]] ... [D.sub.[k.sub.q]]p + ...

= [g.sup.ij][A.sub.ikq][D.sub.j][D.sub.[k.sub.1]] ... [D.sub.[k.sub.q - 1]]p = [[Phi].sup.q].

Estimating the product of tensors in the usual way gives

[absolute value of [x.sup.i][D.sub.i][D.sup.q]p + (q - 1)[D.sup.q]p] [less than or equal to] q [absolute value of A] [absolute value of [D.sup.q]p] + [absolute value of [[Phi].sup.q]].

Applying Corollary 4.3 with T = [D.sup.q]p gives

(q - 1)[absolute value of [D.sup.q]p] [less than or equal to] q [absolute value of A] [absolute value of [D.sup.q]p] + [absolute value of [[Phi].sup.q]].

Now we can make [absolute value of A] [less than or equal to] 1/2 by making r [less than or equal to] c [square root of B] with c small; it is important here that c is independent of q! Then we get

(q - 2)[absolute value of [D.sup.q]p] [less than or equal to] 2[absolute value of [[Phi].sup.q]]

which is a good estimate for q [greater than or equal to] 3. The term [[Phi].sup.q] is estimated inductively from the terms [D.sup.q - 1]p and [D.sup.q - 2]Rm and lower. This proves the following result.

THEOREM 4.6. There exist constants [C.sub.q] for q [greater than or equal to] 3 depending only on q and the dimension and on [absolute value of [D.sup.j]Rm] for j [less than or equal to] q - 2 such that

[absolute value of [D.sup.q]p] [less than or equal to] [C.sub.q]

on the ball r [less than or equal to] c / [square root of [B.sub.0]].

Now we turn our attention to estimating the Euclidean metric [I.sub.jk] and its covariant derivatives with respect to [g.sub.jk]. In this we need a slightly different estimate on our partial differential equation.

THEOREM 4.7. Suppose that f is a function on a ball [absolute value of x] [less than or equal to] r with f(0) = 0

[absolute value of [x.sup.i][Delta]f/[Delta][x.sup.i]] [less than or equal to] C[[absolute value of x].sup.2]

for some constant C. Then

[absolute value of f] [less than or equal to] C[[absolute value of x].sup.2]

for the same constant C.

Proof. Consider the point where [absolute value of f] is largest on the ball [absolute value of x] [less than or equal to] s for s [less than or equal to] r. By switching f with -f if necessary we can assume f [greater than or equal to] 0 there. Rotate so that the point where this occurs is [x.sup.1] = s, [x.sup.2] = ... = [x.sup.n] = 0. Then by the Mean Value Theorem

f(x, 0, ..., 0) = s [Delta]f/[Delta][x.sup.1] ([x.sup.1], 0, ..., 0)

for some [x.sup.1] with 0 [less than or equal to] [x.sup.1] [less than or equal to] s. But by our hypothesis

[x.sup.1] [Delta]f/[Delta][x.sup.1] ([x.sup.1], 0, ..., 0) [less than or equal to] C[([x.sup.1]).sup.2]

so

f(s, 0, ...,0) [less than or equal to] Cs[x.sup.1] [less than or equal to] C[s.sup.2]

and this is enough to prove the theorem.

COROLLARY 4.8. If T = {[T.sub.j...k]} is a tensor which vanishes at the origin and if

[absolute value of [x.sup.i][D.sub.i]T] [less than or equal to] C[[absolute value of x].sup.2]

on a ball [absolute value of x] [less than or equal to] r then [absolute value of T] [less than or equal to] C[[absolute value of x].sup.2] with the same constant C.

Proof. We would like to apply the above theorem to the function f = [absolute value of T]. In case this is not smooth, we use

f = [square root of [[absolute value of T].sup.2] + [[Epsilon].sup.2]] - [Epsilon]]

and let [Epsilon] [right arrow] 0. Note that if T = 0 at the origin then f = 0 also. Also

[absolute value of T] [less than or equal to] f + [Epsilon].

Now we compute

[x.sup.i] [Delta]f/[Delta][x.sup.i] = 1/f + [Epsilon] <T, [x.sup.i][D.sub.i]T>

and hence

[absolute value of [x.sup.i] [Delta]f/[Delta][x.sup.i]] [less than or equal to] C[[absolute value of x].sup.2]

with the same constant C. Then we get f [less than or equal to] C[[absolute value of x].sup.2] for every [Epsilon], and when [Epsilon][right arrow] 0 we get [absolute value of T] [less than or equal to] C[[absolute value of x].sup.2] as desired.

Our application will be to the tensor [I.sub.jk] which gives the Euclidean metric as a tensor in geodesic coordinates. We have

[Mathematical Expression Omitted]

and since

[Mathematical Expression Omitted]

we get the equation

[x.sup.i][D.sub.i][I.sub.jk] = -[g.sup.pq][A.sub.jp][I.sub.kq] - [g.sub.pq][A.sub.kp][I.sub.jq].

We already have [absolute value of [A.sub.jk]] [less than or equal to] [C.sub.0][B.sub.0][[absolute value of x].sup.2] for [absolute value of x] [less than or equal to] r [less than or equal to] c/[square root of [B.sub.0]]. The tensor [I.sub.jk] doesn't vanish at the origin, but the tensor

[h.sub.jk] = [I.sub.jk] - [g.sub.jk]

surely does. We can then use

[x.sup.i][D.sub.i][h.sub.jk] = -[g.sup.pq][A.sub.jp][h.sub.kq] - [g.sup.pq][A.sub.kp][h.sub.jq] - 2[A.sub.jk].

Suppose [absolute value of [h.sub.jk]] [less than or equal to] M(s) on the ball [absolute value of x] [less than or equal to] s. Then

[absolute value of [x.sup.i][D.sub.i][h.sub.jk]] [less than or equal to] 2[1 + M(s)]][C.sub.0][B.sub.0][[absolute value of x].sup.2]

and we get

[absolute value of [h.sub.jk]] [less than or equal to] 2[1 + M(s)][C.sub.0][B.sub.0][[absolute value of x].sup.2]

on [absolute value of x] [less than or equal to] s from Corollary 4.8. This makes

M(s) [less than or equal to] 2[1 + M(s)][C.sub.0][B.sub.0][s.sup.2].

Then for s [less than or equal to] r [less than or equal to] c/[square root of [B.sub.0]] with c small compared to [C.sub.0] we get 2[C.sub.0][C.sub.0][s.sup.2] [less than or equal to] 1/2 and M(s) [less than or equal to]4[C.sub.0][B.sub.0][s.sup.2]. Thus

[absolute value of [h.sub.jk]] [less than or equal to] 4[C.sub.0][B.sub.0][[absolute value of x].sup.2]

for [absolute value of x] [less than or equal to] r. We state this result.

THEOREM 4.9. if [absolute value of x] [less than or equal to] r [less than or equal to] c/[square root of [B.sub.0]]. then

[absolute value of [I.sub.jk] - [g.sub.jk]] [less than or equal to] 4[C.sub.0][B.sub.0][[absolute value of x].sup.2]

and hence for c small enough

1/2[g.sub.jk] [less than or equal to] [I.sub.jk] [less than or equal to] 2[g.sub.jk].

Thus the metrics are comparable. Note that this estimate only needs r small compared to [B.sub.0] and does not need any bounds on the derivatives of the curvature.

Now to obtain bounds on the covariant derivatives of the Euclidean metric [I.sub.kl] with respect to the Riemannian metric [g.sub.kl] we want to start with the equation

[x.sup.i][D.sub.i][I.sub.kl] + [g.sup.mn][A.sub.km][I.sub.ln] + [g.sup.mn][A.sub.lm][I.sub.kn] = 0

and apply q covariant derivatives [D.sub.[j.sub.1]] ... [D.sub.[j.sub.q]]. Each time we do this we must interchange [D.sub.j] and [x.sup.i][D.sub.i], and since this produces a term which helps we should look at it closely. If we write [R.sub.ji] = [[D.sub.j], [D.sub.i]] for the commutator, this operator on tensors involves the curvature but no derivatives. Since

[Mathematical Expression Omitted]

we can compute

[[D.sub.j], [x.sup.i][D.sub.i]] = [D.sub.j] + [g.sup.im][A.sub.jm][D.sub.i] + [x.sup.i][R.sub.ji]

and the term [D.sub.j] in the commutator helps, while [A.sub.jm] can be kept small and [R.sub.ji] is zero order. It follows that we get an equation of the form

[x.sup.i][D.sub.i][D.sub.[j.sub.1]] ... [D.sub.[j.sub.q]][I.sub.kl] + q[D.sub.[j.sub.1]] ... [D.sub.[j.sub.q]][I.sub.kl]

+ [summation of] [g.sup.im][A.sub.jkm][D.sub.[j.sub.1]] where h = 1 to q ... [D.sub.[j.sub.h - 1]][D.sub.i][D.sub.[j.sub.h + 1]] ... [D.sub.[j.sub.q]][I.sub.kl]

+ [g.sup.mn][A.sub.km][D.sub.[j.sub.1]] ... [D.sub.[j.sub.q]][I.sub.ln]

+ [g.sup.mn][A.sub.lm][D.sub.[j.sub.1]] ... [D.sub.[j.sub.q]][I.sub.ln] + [[Psi].sup.q] = 0

where the slush term [[Psi].sup.q] is a polynomial in derivatives of [I.sub.kl] of degree no more than q - 1 and derivatives of p of degree no more than q + 2 (remember [x.sup.i] = [q.sup.ij][D.sub.j]p and [A.sub.ij] = [D.sub.i][D.sub.j]p - [g.sub.ij]) and derivatives of the curvature Rm of degree no more than q - 1. We now estimate

[D.sup.q][I.sub.kl] = {[D.sub.j1] ... [D.sub.[j.sub.q]][I.sub.kl]}

by induction on q using Corollary 4.3 with [Lambda] = q. Noticing a total of q + 2 terms contracting [A.sub.ij] with a derivative of [I.sub.kl] of degree q, we get the estimate

q sup [absolute value of [D.sup.q][I.sub.kl]] [less than or equal to] (q + 2) sup [absolute value of A] sup [absolute value of [D.sup.q][I.sub.kl]] + sup [absolute value of [[Psi].sup.q]].

Now since [absolute value of A] [less than or equal to] [C.sub.0][B.sub.0][r.sup.2] and r [less than or equal to] c/[square root of [B.sub.0]], if we take c sufficiently small so that sup [absolute value of A] [less than or equal to] 1/4, then for all q [greater than or equal to] 1

(3q - 2) sup [absolute value of [D.sup.q][I.sub.kl]] [less than or equal to] 4 sup [absolute value of [[Psi].sup.q]]

and everything works. This proves the following result.

THEOREM 4.10. There exists a constant c [greater than] 0 depending only on the dimension, and constants [C.sub.q] depending only on the dimension and q and bounds [B.sub.j] on the curvature and its derivatives for j [less than or equal to] q where [absolute value of [D.sub.j]Rm] [less than or equal to] [B.sub.j], so that for any metric [g.sub.kl] in geodesic coordinates in the ball [absolute value of x] [less than or equal to] r [less than or equal to] c/[square root of [B.sub.0]] the Euclidean metric [I.sub.kl] satisfies

1/2[g.sub.kl] [less than or equal to] [I.sub.kl] [less than or equal to] 2[g.sub.kl]

and the covariant derivatives of [I.sub.kl] with respect to [g.sub.kl] satisfy

[absolute value of [D.sub.j1] ... [D.sub.[j.sub.q]][I.sub.kl]] [less than or equal to] [C.sub.q].

COROLLARY 4.11. We also have estimates

1/2[I.sub.kl] [less than or equal to] [g.sub.kl] [less than or equal to] 2[I.sub.kl]

and

[Mathematical Expression Omitted]

for similar constants [Mathematical Expression Omitted].

Proof. The difference between a covariant derivative and an ordinary derivative is given by the connection [Mathematical Expression Omitted], which we can recover from solving

[Mathematical Expression Omitted]

to get

[Mathematical Expression Omitted].

This gives us bounds on [Mathematical Expression Omitted]. We then obtain bounds on the first derivatives of [g.sub.ij] from

[Mathematical Expression Omitted].

Now we get bounds on covariant derivatives of [Mathematical Expression Omitted] from the covariant derivatives of [I.sub.pk] and bounds on the ordinary derivatives of [Mathematical Expression Omitted] by relating them to the covariant derivatives using the [Mathematical Expression Omitted], and bounds on the ordinary derivatives of the [g.sub.jk] from bounds on the ordinary derivatives of the [Mathematical Expression Omitted], always proceeding inductively on the order of the derivative. This proves the corollary.

5. In this section we show how to estimate the derivatives of an isometry.

THEOREM 5.1. Let y = F(x) be an isometry from a ball in Euclidean space with a metric [g.sub.ij]d[x.sup.i]d[x.sup.j] to a ball in Euclidean space with a metric [h.sub.pq]d[y.sup.p]d[y.sup.q]. Then we can bound all of the derivatives of y with respect to x in terms of bounds on the derivatives of [g.sub.ij] with respect to x and bounds on the derivatives of [h.sub.pq] with respect to y.

Proof. Since y = F(x) is an isometry we have the equation

[h.sub.pq] [Delta][y.sup.p]/[Delta][x.sup.j] [Delta][y.sup.q]/[Delta][x.sup.k] = [g.sub.jk].

Using bounds [g.sub.ij] [less than or equal to] C[I.sub.jk] and [h.sub.pq] [greater than or equal to] c[I.sub.pq] comparing to the Euclidean metric, we easily get estimates

[absolute value of [Delta][y.sup.p]/[Delta][x.sup.j]] [less than or equal to] C.

Now if we differentiate the equation with respect to [x.sup.i] we get

[h.sub.pq][[Delta].sup.2][y.sup.p]/[Delta][x.sup.i][Delta][x.sup.j] [Delta][y.sup.q]/[Delta][x.sup.k] + [h.sub.pq][Delta][y.sup.p]/[Delta][x.sup.j] [[Delta].sup.2][y.sup.q]/[Delta][x.sup.i][Delta][x.sup.k]

= [Delta][g.sub.jk]/[Delta][x.sup.i] - [Delta][h.sub.pq]/[Delta][y.sup.r] [Delta][y.sup.r]/[Delta][x.sup.i] [Delta][y.sup.p]/[Delta][x.sup.j] [Delta][y.sup.q]/[Delta][x.sup.k].

Now let

[T.sub.ijk] = [h.sub.pq][Delta][y.sup.p]/[Delta][x.sup.i] [[Delta].sup.2][y.sup.q]/[Delta][x.sup.j][Delta][x.sup.k]

and let

[U.sub.ijk] = [Delta][g.sub.jk]/[Delta][x.sup.i] - [Delta][h.sub.pq][Delta][y.sup.r] [Delta][y.sup.r]/[Delta][x.sup.i] [Delta][y.sup.p]/[Delta][x.sup.j] [Delta][y.sup.q]/[Delta][x.sup.k].

Then the above equation says

[T.sub.kij] + [T.sub.jik] = [U.sub.ijk].

Using the obvious symmetries [T.sub.ijk] = [T.sub.ikj] and [U.sub.ijk] = [U.sub.ikj] we can solve this in the usual way to obtain

[T.sub.ijk] = 1/2([U.sub.jik] + [U.sub.kij] - [U.sub.ijk]).

We can recover the second derivatives of y with respect to x from the formula

[[Delta].sup.2][y.sup.p]/[Delta][x.sup.i][Delta][x.sup.j] = [g.sup.kl][T.sub.kij][Delta][y.sup.p]/[Delta][x.sup.l].

Combining these gives an explicit formula giving [[Delta].sup.2][y.sup.p]/[Delta][x.sup.i][x.sup.j] as a function of [g.sub.ij], [h.sub.pq], [Delta][g.sub.jk]/[Delta][x.sup.i], [Delta][h.sub.pq]/[Delta][y.sup.r], and [Delta][y.sup.p]/[Delta][y.sup.i]. This gives bounds

[absolute value of [[Delta].sup.2][y.sup.p]/[Delta][y.sup.i][Delta][y.sup.j]] [less than or equal to] C

and bounds on all higher derivatives follow by differentiating the formula and using induction.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, SAN DIEGO, LA JOLLA, CA 92093-0112

REFERENCES

[CG&T] J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15-53.

[F1] K. Fukaya, Collapsing Riemannian manifolds to ones of lower dimension, J. Differential Geom. 25 (1987), 139-156.

[F2] -----. J. Differential Geom. (1989).

[G&W] R. Greene and H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988), 119-141.

[G] M. Gromov, Structures Metriques pour les Varietes Riemanniennes, Textes Math., vol. 1, Cedric-Nathan, Paris, 1981.

[G&K] K. Grove and H. Karcher, How to conjugate [C.sup.1] close group actions, Math. Z. 132 (1973), 11-20.

[P1] S. Peters, Cheeger's finiteness theorem for diffeomorphism classes of Riemannian manifolds, J. Reine Angew. Math. 349 (1984), 77-82.

[P2] -----, Convergence of Riemannian manifolds, Comp. Math. 62 (1987), 3-16.

[S] W. X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), 223-301; Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989), 303-394.

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Author: | Hamilton, Richard S. |
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Publication: | American Journal of Mathematics |

Date: | Jun 1, 1995 |

Words: | 12651 |

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