c Birkhäuser Verlag, Basel 1997 GAFA Geometric And Functional Analysis

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1 GAFA, Geom. funct. anal. Vol. 7 (997) X/97/63-5 $.5+.2/ c Birkhäuser Verlag, Basel 997 GAFA Geometric And Functional Analysis RELATIVE VOLUE COPARISON WITH INTEGRAL CURVATURE BOUNDS P. Petersen and G. Wei Abstract In this paper we shall generalize the Bishop-Gromov relative volume comparison estimate to a situation where one only has an integral bound for the part of the Ricci curvature which lies below a given number. This will yield several compactness and pinching theorems. Introduction In this paper we shall be concerned with proving some results related to the work from [PeSW]. The techniques here are similar, but independent of the developments in [PeSW], still we suggest that the reader read at least the introduction to [PeSW]. Also our techniques are different from those used in [G] and [Y]. This paper can therefore be read without prior knowledge of those papers. We will present a new relative volume comparison estimate which generalizes the classical Bishop-Gromov comparison inequality. The consequences of this are manifold and hopefully far reaching. To state our results we need some notation. On a Riemannian manifold define the function g : [, ) asg(x) = the smallest eigenvalue for Ric : T x T x. Now consider ( ) pd k (λ, p) = max { g (x)+(n ) λ, } vol k (λ, p) = ( ) pd max { g (x)+(n ) λ, } vol, vol the last quantity is the averaged amount of curvature below (n ) λ. This quantity is in many ways more natural than the first. We of course have that Ric () (n ) λ iff k (λ, p) =. Our main result is The first author was supported in part by NSF and NYI grants and the second author was supported in part by an NSF grant.

2 32 P. PETERSEN AND G. WEI GAFA Theorem.. Let x, λ, andp>n/2be given, then there is a constant C (n, p, λ, R) which is nondecreasing in R such that when r<r we have ( ) vol B(x, R) / ( ) vol B(x, r) / C(n, p, λ, R) (k(λ, p)) /. v(n, λ, R) v(n, λ, r) Furthermore when r =we obtain vol B (x, R) ( +C(n, p, λ, R) (k (λ, p)) / ) v (n, λ, R). Note that when Ric (n ) λ, i.e. k (p, λ) =, this gives the classical relative volume comparison estimate. Our proof of this volume estimate does not use the setup used in Gallot and Yang s work. In fact our proof is somewhat different and is more inspired by some of the new estimates obtained in [PeSW]. The absolute volume estimate is actually better than the one in [Y], as we recover the correct volume estimates when k (λ, p) =. In [Y] one only arrives at the correct volume bound when k(,p)=. As a corollary we obtain the following volume doubling result: Corollary.2. Under the same conditions as the above theorem we have that for all α<there is an ε = ε(n, p, λ, D, α) > such that if is a Riemannian manifold with diam D and k (λ, p) ε, then for all x and r<dwe have v (n, λ, r) vol B (x, r) α v (n, λ, D) vol. As an immediate consequence we have the following extension of Gromov s precompactness result: Corollary.3. Given an integer n>,p>n/2and λ, D<,we can find ε (n, p, λ, D) such that the class of closed Riemannian n-manifolds with diam D, k(λ, p) ε is precompact in the Gromov-Hausdorff topology. The classical relative volume comparison result has proven very useful in many contexts. So one would expect the above inequality to give some new results that are similar but more general. Here we shall concentrate on compactness results. In a future paper we will show how other finiteness result also generalize when we use our new volume estimate. The relative volume estimate will enable us to obtain some very general compactness and pinching results, where in addition to assuming lower

3 Vol. 7, 997 RELATIVE VOLUE COPARISON 33 volume bounds and upper diameter bounds one has some sort of L p curvature bounds. The reader who is unfamiliar with the standard language of convergence theory might wish to consult [Pe] for a complete survey of results and proofs of this area of geometry. Theorem.4. Given an integer n 2, andnumbersp>n/2,λ, v>,d<,λ<, one can find ε = ε (n, p, λ, D) > such that the class of closed Riemannian n-manifolds with vol vol () v diam () D R p Λ k (λ, p) ε (n, p, λ, D) is precompact in the C α, α<2 n p, topology. Note that ε in the previous theorem does not depend on Λ or v. The smallness condition on k (λ, p) is inevitable as is shown by examples in both [G] and [Y]. In fact in [Y] there are examples which have arbitrarily high Betti numbers but satisfy vol () v diam () D R p Λ for some constants v, D, Λ. Recall that a Riemannian metric (,g) has constant sectional curvature λ iff the Riemannian curvature (, 4)-tensor R = λg g, where g g is the Kulkarni-Nomizu product. Corollary.5. Given an integer n 2, andnumbersp>n/2,λ R, v >, D <, one can find ε = ε (n, p, λ, D) > such that a closed Riemannian n-manifold (,g) with vol () v, diam () D, R λ g g p ε (n, p, λ, D) vol is C α, α<2 n p close to a constant curvature metric on.

4 34 P. PETERSEN AND G. WEI GAFA Corollary.6. Given an integer n 2, andnumbersp>n/2,λ R, v>,andλ,d <, one can find ε = ε (n, p, λ, D) > such that a closed Riemannian n-manifold (,g) with vol () v, diam () D, R p Λ, Ric λ (n ) g p ε (n, p, λ, D) vol is C α, α<2 n p close to an Einstein metric on. These two pinching results, and the compactness theorem as well, are to our knowledge the most general of their type. Namely, where, aside from curvature conditions, one only has global diameter and volume bounds. In section 3 we shall also discuss how one gets compactness theorems for complete manifolds, and also how one can generalize the curvature condition R p Λ to a slightly different curvature condition. This will lead us to a very attractive Ricci curvature pinching theorem. Note also that we can in all of the above results replace the two conditions vol () v, R p Λ, by the single condition inj () i, as is done in [A]. Along these lines we can also study manifolds with almost maximal volume as is also done in [A]. Specifically we have Theorem.7. Given an integer n 2, andnumbersp>n/2,λ, r>,λ<, one can find ε = ε(n, p, λ r 2 ) > and δ = δ(n, λ r 2 ) > such that the class of complete Riemannian n-manifolds with vol B (x, r) ( δ) v (n, λ, r) for all x, Ric p Λ, k (λ, p) ε is precompact in the C α, α<2 n p, topology.

5 Vol. 7, 997 RELATIVE VOLUE COPARISON 35 Note that ε in the previous theorem does not depend on Λ and that δ depends only on the dimension. As an immediate corollary we have a very nice L p Ricci pinching result Corollary.8. Given an integer n 2, andnumbersp>n/2,λ R, r > and Λ,D <, one can find ε = ε (n, p, λ, r, D) > and δ = δ(n, λ D 2 ) > such that a closed Riemannian n-manifold (,g) with vol B (x, r) ( δ) v (n, λ, r) for all x, diam () D, Ric (n ) λ g p ε is C α, α<2 n p close to an Einstein metric on. In the last section of this paper we also discuss how some of the L n/2 compactness and pinching results of Gao and Anderson fit into this new framework. Acknowledgments. The second author would like to thank UCLA for their hospitality during her stay there. The first author would like to thank Robert Brooks for the many discussions we have had on volume comparison with integral curvature bounds. The authors would also like to thank the referee for making several useful suggestions and corrections. 2 The Volume Estimate We suppose that we are given a complete Riemannian n-manifold and x. Around x use exponential polar coordinates and write the volume element as d vol = ωdt dθ n,wheredθ n is the standard volume element on the unit sphere S n (). As t increases ω becomes undefined but we can just declare it to be zero for those t. We know that ω = hω, whereh is the mean curvature of the distance spheres around x, and hsatisfies the differential inequality h + h2 n Ric ( t, t ). Here t is the unit gradient of the distance function d (,x). In the n-dimensional space form Sλ n of constant curvature λ, wecan similarly choose y Sλ n and write the volume element as d vol = ω λdt dθ n.weuseλ in order that the polar coordinates are defined on all of Sλ n {y}and also so that h λ is non-negative everywhere. Now we have

6 36 P. PETERSEN AND G. WEI GAFA again ω λ = h λω λ where this time the mean curvature satisfies Thus we get that h λ + h2 λ = (n ) λ. n h λ (t) =h λ (t, ) =(n ) sn λ (t) sn λ (t), ω λ (t) =ω λ (t, ) = sn n λ (t), where sn λ is the unique solution to ϕ + λϕ =, ϕ() =, ϕ () =. Having similar coordinate systems on and Sλ n allows us to compare their volume forms and mean curvatures. Let us define ψ = ψ (t, ) = max {,h(t, ) h λ (t, )} and declare that ψ is whenever it becomes undefined. Lemma 2.. For the volume ratio we have that d vol B (x, r) dr v (n, λ, r) C (n, λ, r) vol v(n,λ,r) ( ) vol B (x, r) ( ) ψ d vol (v (n, λ, r)), v (n, λ, r) where t ω λ (t) C (n, λ, r) = max t t [,r] ω λ (s) ds, C (n, λ, ) = n. Proof. First observe that the fraction ω/ω λ satisfies d ω (h h λ ) ω dt ω λ ω λ ψ ω ω λ. Note that away from the cut locus the first inequality is actually an equality. At the cut locus the singular part of the derivative of ω has negative measure, hence when the derivative is interpreted correctly we get inequality. This implies d ω (r, ) dθ n Sn = dr S ω n λ (r) dθ n vol S n S n d ω (r, ) dr ω λ (r) dθ n

7 Vol. 7, 997 RELATIVE VOLUE COPARISON 37 Thus for t r we have ω (r, ) dθ n Sn S ω n λ (r, ) dθ n vol S n ψ ω dθ n. S n ω λ Sn ω (t, ) dθ n S ω n λ (t, ) dθ n r vol S n ψ ω dθ n ds, t S n ω λ which implies ω(r, )dθ n ω λ (t)dθ n ω λ (r)dθ n ω(t, )dθ n S n S n S n S ( )( ) n vol S n ω λ (r) dθ n ω λ (t) dθ n S n S n r ψ ω dθ n ds t S n ω ( λ ) r ω λ (r) dθ n ψωdθ n ds S n t S n r ω λ (r)vols n ψωdθ n ds S ( n r ) / ω λ (r)vols n ψ ωdθ n ds S n ( r ) ωdθ n ds S n vol S n ω λ (r)(volb(x, r)) ( ψ d vol ) /. Using the volume elements from above we can write r vol B (x, r) v (n, λ, r) = S ω dt S n ω n λ dθ n dt. r Thus we have ( ) ( r d vol B (x, r) dr v (n, λ, r) = S ω (r, ) dθ n n S ω n λ (t) dθ n dt ) (v (n, λ, r)) 2 ( S ω (t, ) dθ n n dt ) ) ( r S ω n λ (r) dθ n (v (n, λ, r)) 2. Now observe that the numerator can be written as r ( ) ω (r, ) dθ n ω λ (t) dθ n dt S n S n

8 38 P. PETERSEN AND G. WEI GAFA r ( ω λ (r) dθ n ω (t, ) dθ n S n S n r ( ( vol S n ω λ (r) (vol B (x, r)) =vols n r ω λ (r) (vol B (x, r)) ( Thus we have d vol B (x, r) dr v (n, λ, r) vol Sn r ω λ (r) (vol B (x, r)) (v (n, λ, r)) 2 ( ) vol B (x, r) ( C (n, λ, r) ψ d vol v (n, λ, r) Here we used that vol S n r ω λ (r) v (n, λ, r) ) dt = r ω λ (r) r ω λ (s) ds ) / ) ψ d vol dt ψ d vol ) /. ( ψ d vol ) ) (v (n, λ, r)). converges to n as r and can therefore be estimated by its maximum value C (n, λ, r) on[,r]. Note that when λ< the constant C if either r or λ. On the other hand when λ =wecanuse C =nfor all r. We must now estimate ψ d vol in terms of k (p, λ). This estimate is inspired by our similar estimates in [PeSW, Section 3]. To reduce the problem first write r ψ d vol = ψ ωdt dθ n. S n Thus it suffices to estimate r ψ ωdt. Now define ρ = ρ(t, ) = max{, (n )λ Ric( t, t )}. If we are beyond where the coordinate system is defined then we use the default that ρ is zero. Also define r k (p, λ, r) = ρ p d vol = (ρ (t, )) p ωdt dθ n S n k(p, λ, r) k (p, λ, r) = sup x vol B (x, r). With this notation behind us it is now clear that d dr vol B (x, r) v (n, λ, r) C 3 (n, p, λ, r) ( ) vol B (x, r) (k (p, λ, r)) (v (n, λ, r)). v (n, λ, r)

9 Vol. 7, 997 RELATIVE VOLUE COPARISON 39 provided we can prove Lemma 2.2. There is a constant C 2 (n, p) such that when p>n/2we have r ψ ωdt C 2 (n, p) r ρ p ωdt. Proof. We have that ψis absolutely continuous and satisfies ψ + +2ψ h λ n n ρ. ultiply through by ψ 2 ω and integrate to get r ψ ψ 2 ωdt + r ψ ωdt + 2 r h λ ψ ωdt n n Integration by parts yields r ψ2 r ρ ψ 2 ωdt. ψ ψ 2 ωdt r = ψ ω r ψ hωdt r ψ hωdt r ψ ωdt r ψ h λ ωdt. Inserting this in the above inequality we obtain ( n ) r ( 2 ψ ωdt + n ) r h λ ψ ωdt r ρ ψ 2 ωdt. When p>n/2 we therefore obtain ( n ) r r ψ ωdt ρ ψ 2 ωdt ( r ) ( ρ p p r ) ωdt ψ p ωdt. By dividing through by ( r ψ ωdt ) p we then get ( n )( r /p ( r /p ψ ωdt) ρ ωdt) p. Or in other words ( r ) ( ψ ωdt n ) p ( r ) ρ p ωdt,

10 4 P. PETERSEN AND G. WEI GAFA with C 2 (n, p) = ( n ) p. We therefore arrive at the desired inequality. We can now prove Lemma 2.3. There is a constant C(n, p, λ, R) which is nondecreasing in R such that when r<rwe have ( ) vol B (x, R) / ( ) vol B (x, r) / C (n, p, λ, R) (k (p, λ, R)) /. v (n, λ, R) v (n, λ, r) Furthermore, when r =we obtain vol B (x, R) ( +C(n, p, λ, R) (k (p, λ, R)) / ) v (n, λ, R). Proof. From Lemma 2. we have a differential inequality of the type y α y f(x), y () = and y>. Separation of variables and integration yields So we can simply use y / (R) y / (r) α C = C 3 R R (v (n, λ, t)) / dt. r f (x) dx. Furthermore, we can use the initial condition y ()=toget R y / (R)+ α f(x)dx. The final observation to be made is that the integral R (v (n, λ, t)) / dt indeed converges when p>n/2. To see this recall that v (n, λ, t) t n as t. So the integrand looks like t n/ as t. And this function is integrable when n/ >, or in other words when p>n/2. Observe that the condition p>n/2isnecessary both to get the estimate for ψ ωdt and for the integrability of (v (n, λ, t)) /. It seems lucky indeed that those two different conditions are the same. Finally we can use the inequality k (p, λ, r) k (p, λ) to obtain the relative volume comparison estimate mentioned in the introduction. Another remark is in order at this point. We never really seriously use that λ, and indeed one can extend the estimates to hold for λ>.

11 Vol. 7, 997 RELATIVE VOLUE COPARISON 4 However, if R or r becomes larger than π / λ then we will run into trouble. Also there are a few things to worry about when r>π / 2 λas h λ becomes negative there. We can now show Corollary 2.4. Given an integer n>and p>n/2,λ,andd>, we can for all α<find ε = ε (n, p, λ, D, α) > such that any complete Riemannian n-manifold with diam D and k(p, λ) ε satisfies for all x and r<d v(n, λ, r) vol B (x, r) α v (n, λ, D) vol. Proof. Consider the inequality ( ) vol / ( ) vol B (x, r) / C (n, p, λ, D) (k (p, λ)) / v (n, λ, D) v (n, λ, r) and then cross multiply to get ( ) v (n, λ, r) / ( vol B(x, r) v (n, λ, D) vol ) / C (n, p, λ, D) (v (n, λ, r)) / ( ) k (p, λ) /. vol Now choose ε so that (C (n, p, λ, D)) ε ( α) v (n, λ, D). Then we have that ( ) v (n, λ, r) / ( vol B (x, r) v (n, λ, D) vol as long as k (p, λ) = This is the desired estimate. ) / ( ) v (n, λ, r) / ( α) v (n, λ, D) k(p, λ) vol ε. 3 Pinching and Compactness In the survey article [Pe] it is proven that all the corollaries in the introduction are immediate consequences of the compactness theorems we mention there. Observe that the compactness theorems can be applied after one gets smallness for k (p, λ) from k (p, λ ) Ric λ (n ) g p. vol

12 42 P. PETERSEN AND G. WEI GAFA The compactness theorems are also almost immediate consequences of the results in [A], [Y] and [Pe]. Namely, from the last two sources it follows that for any v, r,d,λ (, ) the class satisfying vol B (x, r) v r n for r r, diam () D, R p Λ is precompact in the C α, α<2 n p, topology. Thus we must establish this local volume growth condition from the conditions vol v, diam D, k(p, λ) ε. Using α =/2 in Corollary 2.4 we can clearly find ε (n, p, λ, D) such that the condition k (p, λ) ε implies v (n, λ, r) vol B (x, r) 2 v (n, λ, D) vol. Using the volume estimate vol v we then obtain vol B (x, r) 2 v (n, λ, r) v v (n, λ, D) C (n, p, λ, v, D) r n for r D. It should be observed that we also get a compactness theorem for complete manifolds (see e.g. [Pe]). Theorem 3.. Given an integer n 2, andnumbersp>n/2,λ, r>,v>,λ<, one can find ε = ε (n, p, λ, r) > such that the class of complete Riemannian n-manifolds with vol B (x, r) v for all x, R p Λ for all x, k(λ, p, r) ε (n, p, λ, r) is precompact in the C α, α<2 n p, topology as well. In [Pe] it is discussed how one can replace the R p Λbythe

13 Vol. 7, 997 RELATIVE VOLUE COPARISON 43 two conditions Ric p Λ, R n/2 ɛ, where ɛ is some small number depending on dimension r and the lower volume bound v for the balls B (x, r). These are actually the conditions used in Yang s work. Using this we can re-state all of the above compactness and pinching theorems with the condition Ric p Λ, R n/2 ɛ. replacing R p Λ. This gives us a particularly nice pinching result. Theorem 3.2. Given an integer n 2 and numbers p>n/2,λ R, r>,v>,d<, we can find ε (n, p, λ, D) > and ɛ (n, λ, r, v) > such that any closed Riemannian manifold (,g) satisfying diam () D, vol B (x, r) v for all x, R n/2 ɛ for all x, Ric (n ) λ g p ε for all x vol is C α, α<2 n p close to an Einstein metric on. In [A, Theorem 2.6] and [AC] the authors study the situation when R n/2 is merely bounded rather than small. The convergence and finiteness results obtained there clearly also admit generalization to the situation where, instead of having Ric Λ, we assume Ric p Λ and smallness of k (p, λ, r). The compactness and pinching results that use almost maximal volume hypotheses rest on the following key result proved in [A, Gap Lemma 3.]. Theorem 3.3. There is an η (n) > such if is a complete Ricci flat Riemannian n-manifold with the property that vol B (p, r) ( η) v (n,,r) for all r> and some p, thenis isometric to R n.

14 44 P. PETERSEN AND G. WEI GAFA We now need to observe that if we rescale manifolds in the class vol B (x, r) ( δ) v (n, λ, r) for all x, Ric p Λ, k (λ, p, r) ε (n, p, λ, r), then we get complete Ricci flat manifolds. The needed volume growth condition comes from our volume assumption together with the smallness of k (p, λ) in the following way. We have assumed that vol B (x, r) ( δ) v (n, λ, r) for all x, k(λ, p, r) ε (n, p, λ, r) for some fixed r. Ifs<rthen our relative volume comparison result tells us that vol B (x, r) δ v (n, λ, r) (( ) vol B (x, s) / ) + C (n, p, λ, r) ε / v (n, λ, s) ( ( ) vol B (x, s) vol B (x, s) / ) v (n, λ, s) + + C(n, p, λ, r) ε / v (n, λ, s) vol B (x, s) v (n, λ, s) +3 C(n, p, λ, r) ε /, whereweassumedthatc(n, p, λ, r) ε / for the penultimate inequality and that (vol B (x, s) ) / +C(n, p, λ, r) ε / 2 v (n, λ, s) for the last inequality. We can now choose δ (n) = η(n) 2 and in addition ε = ε (n, p, λ, r) such that 3 C (n, p, λ, r) ε / η (n). 2 With these choices for ε and δ we get that, for all s r, vol B(x, s) η v(n, λ, s). Therefore, if we rescale the metrics in the class by constants which go to infinity we obtain metrics that satisfy the hypotheses in Anderson s Gap Lemma.

15 Vol. 7, 997 RELATIVE VOLUE COPARISON 45 We also get some new results on Sobolev constants. Recall that if we have an n-dimensional Riemannian manifold (, g) thenforq [n, ] we can define C q vol n H () =inf, (vol n Ω) q where H runs over all closed hypersurfaces in dividing into two pieces and Ω is the piece with the smallest volume. In [G] it was shown that when q>none can bound C q () intermsvol, diam, andk(q/2,λ). We obviously can t improve this to cover the case where q = n, however using the method in [Y] we obtain bounds for the classical Sobolev constant C n () intermsofvol,diam, and k(p, λ), p > n/2. One should compare this to the Sobolev constant estimates obtained in [BPPe] where the authors instead of assuming a diameter bound consider the spectrum. References [A].T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invt. ath. 2 (99), [AC].T. Anderson, J. Cheeger, Diffeomorphism finiteness for manifolds with Ricci curvature and L n/2 curvature bounded, GAFA (99), [BPPe] R. Brooks, P. Perry, P. Petersen, Spectral geometry in dimension 3, Acta ath. 73 (994), [G] S. Gallot, Isoperimetric inequalities based on integral norms of Ricci curvature, Astérisque (988), [Pe] P. Petersen, Convergence theorems in Riemannian geometry, in Comparison Geometry (K. Grove, P. Petersen, eds.), SRI Publications vol. 3 (997), Cambridge Univ. Press, [PeSW] P. Petersen, S. Shteingold, G. Wei, Comparison geometry with integral curvature bounds, in this issue. [Y] D. Yang, Convergence of Riemannian manifolds with integral bounds on curvature I, Ann. Sci. Écol. Norm. Sup. 25 (992), Peter Petersen Guofang Wei Department of athematics Department of athematics University of California University of California Los Angeles, CA 995 Santa Barbara, CA petersen@math.ucla.edu wei@math.ucsb.edu Submitted: August 996 Revision: July 997