PINCHING THEOREM FOR THE VOLUME ENTROPY FRANÇOIS LEDRAPPIER AND XIAODONG WANG 1. Intoduction Fo a compact Riemannian manifold (M n ; g) with Ric (g) (n 1), we have the following shap geometic inequalities fo which the equality chaacteizes the standad sphee (S n ; g 0 ) in each case: (1) (Bishop-Gomov) the volume vol (M; g) vol (S n ; g 0 ); () (Myes-Cheng) the diamete diam (M; g) ; (3) (Lichneovicz-Obata) the st eigenvalue 1 (M; g) n. Fo the st inequality, Colding [Co1] poved that it is stable. Theoem 1. (Colding) Thee exists a constant " n s.t. if a compact Riemannian manifold (M n ; g) satis es Ric (g) (n 1) and vol (M; g) > vol (S n ; g 0 ) ", then M is Gomov-Hausdo close to S n in the sense that d GH (M; S n ) (")! 0 as "! 0. By Cheege-Colding [ChC3], M is di eomophic to S n if they ae Gomov- Hausdo close. The othe two inequalities tun out to be non-stable. In each case, one obtains a stable vesion if a stonge invaiant is used instead. Regading the second inequality, the diamete should be eplaced by the adius which is de ned to be ad (M; g) = inf sup d ( y) : ym xx Theoem. (Colding [Co]) Thee exists a constant " n s.t. if a compact Riemannian manifold (M n ; g) satis es Ric (g) (n 1) and ad (M; g) > ", then M is Gomov-Hausdo close to S n. Regading the thid inequality, Petesen [P] poved the following Theoem 3. Thee exists a constant " n s.t. if a compact Riemannian manifold (M n ; g) satis es Ric (g) (n 1) and n+1 (M; g) > n ", then M is Gomov- Hausdo close to S n. Moe ecently, Auby [A] poved that the esult emains valid if n+1 is eplaced by n. We ae inteested in compact Riemannian manifold (M n ; g) with Ric (g) (n 1). Notice that any metic can be scaled to satisfy this cuvatue assumption. Moeove, by the wok of Lohkamp [L] that any compact manifold M n with n 3 The st autho was patially suppoted by NSF gant DMS-080117. The second autho was patially suppoted by NSF gant DMS-0905904. 1
FRANÇOIS LEDRAPPIER AND XIAODONG WANG admits metics with negative Ricci cuvatue. Nevetheless, thee ae two natual geometic inequalities which chaacteize hypebolic manifolds in the equality case. Fist, we need to intoduce two invaiants. Let : M f! M be the univesal coveing. Let 0 be the in mum of the L spectum of M, f i.e. 0 = inf Rf M juj Rf M u ; whee the in mum is taken ove all smooth functions with compact suppot. The volume entopy v is de ned by v = lim!1 ln volb ( ) ; whee B ( ) is the ball of adius centeed at x in f M. It is well-known that 0 v =4. Theoem 4. Let M n be a compact Riemannian manifold with Ric (n 1). Then 0 (n 1) =4 and equality holds i M is hypebolic. Theoem 5. Let M n be a compact Riemannian manifold with Ric (n 1). Then the volume entopy satis es v n 1 and equality holds i M is hypebolic. Theoem 4 was poved by the second autho [W] using the Kaimanovich entopy. Theoem 5, which implies Theoem 4 in view of the well-known fact 0 v =4, was ecently poved by the authos [LW]. A natual question is whethe these two inequalities ae stable. In fact, pio to ou wok [LW] Theoem 5 had been known unde the additional condition that M is negatively cuved as a theoem of Kniepe [Kn]. In an unpublished manuscipt in 000, Coutois took up the stability question and poved the following Theoem 6. Thee exists a positive constant " = " (n; D) s.t. if (M n ; g) is a compact Riemannian manifold of dimension n satisfying the following conditions g has negative sectional cuvatue, Ric (g) (n 1), diam (M; g) D, the volume entopy v (g) n 1 ", then M is di eomophic to a hypebolic manifold (X; g 0 ). Moeove, the Gomov- Hausdo distance d GH (M; X) (")! 0 as "! 0. His poof is based on the theoy of Cheege-Colding [ChC] on almost igidity. The pupose of this pape is to pesent a di eent appoach based on ou pevious wok. We show that the method we developed in [LW] to pove the igidity theoem can be stengthen to given the following igidity theoem fo C 1; metics. Theoem 7. Let M n be a (smooth) compact manifold and g a C 1; metic. Suppose that g i is a sequence of Riemannian metics on M s.t. (1) Ric (g i ) (n 1) fo each i, () g i! g in C 1; nom as i! 1, (3) the volume entopy v (g i )! n 1 as i! 1. Then g is hypebolic. Fom this igidity esult, we can deduce the Theoem of Coutois in a simple way.
PINCHING THEOREM FOR THE VOLUME ENTROPY 3. Poof of Theoem 7 We st indicate that some of the esults in ou pevious pape [LW] ae valid fo a C 1; Riemannian metic. Let M n be a compact smooth manifold with a C 1; Riemannian metic g. Fix a point o M f and de ne, fo x M f the function x (z) on M f by: x (z) = d( z) d( o): The assignment x 7! x is continuous, one-to-one and takes values in a elatively compact set of functions fo the topology of unifom convegence on compact subsets of M. f The Busemann compacti cation M c of M f is the closue of M f fo that topology. The space M c is a compact sepaable space. The Busemann bounday @ M c := M c nm f is made of Lipschitz continuous functions on M f such that (o) = 0. Elements of @ M c ae called hoofunctions. To each point M c is associated the pojection W of M f fg. As a subgoup of G, the stabilize G of the point acts discetely on M f and the space W is homeomophic to the quotient of M f by G. We put on each W the smooth stuctue and the metic inheited fom. The manifold W and its metic vay continuously on X M. The collection of all W ; M c fom a continuous lamination W M with leaves which ae manifolds locally modeled on M. f In paticula, it makes sense to di eentiate along the leaves of the lamination and we denote W and div W the associated gadient and divegence opeatos: W acts on continuous functions which ae C 1 along the leaves of W, div W on continuous vecto elds in T W which ae of class C 1 along the leaves of W. n On the Busemann bounday @ M c we can constuct a family of nite measues x : x M f o s.t. (1) Fo any pai y, the two measues x and y ae equivalent with the Radon-Nikodium deivative d x () = e v((x) d y (y)) ; () fo any x = x : Then the measue = e v(x) d o () dx is G-invaiant on f M c M and hence descends to a nite measue on X M. By scaling we assume to be a pobability measue. It is then poved that fo all W vecto eld Y which is C 1 along the leaves and globally continuous, (.1) div W Y d = v Y; W d: Since g is C 1;, the heat kenel p t ( y) on f M is C ;. As in [LW], we apply the fomula to the following vecto eld on X M Y t ( ) = (P t ) (x) :
4 FRANÇOIS LEDRAPPIER AND XIAODONG WANG We now cove M by nitely many open sets fu i : 1 i kg s.t. each U i is so small that 1 (U i ) is the disjoint union of open sets each di eomophic to U i via. Let f i g be a patition of unity subodinating to fu i g. Fo each U i let U e i be one of the components of 1 (U i ) and let e i be the lifting of i to U e i. By the same agument we aive at the following fomula: fo any kx (.) e v(x) ( e i ) dx d o () = 0: @ M c U e i i=1 We now futhe assume that thee is a sequence of smooth metics g i on M s.t. Ric (g i ) (n 1) fo each i, g i! g in C 1; nom as i! 1. Lemma 1. We have e (n 1) 0 in the sense of distibution, i.e. fo any Cc 1 with 0 e (n 1)(x) (x) dv (x) 0: Poof. We will denote by i ; d i the Laplacian and distance function on f M w..t. the metic g i. By the Laplacian compaison theoem, fo any a f M we have i d i ( a) (n 1) cosh (d i ( a)) sinh (d i ( a)) in the distibution sense, i.e. fo any Cc 1 with 0 cosh (d i ( a)) d i ( a) i (x) dv i (x) (n 1) sinh (d i ( a)) (x) dv i (x) : Taking limit yields (.3) d ( a) (x) dv (x) (n 1) cosh (d ( a)) (x) dv (x) : sinh (d ( a)) Let @ c M. Then thee exists a sequence fa k g f M s.t. d (o; a k )! 1 and (x) = lim k!1 d ( a k) d (o; a k ) : hee the convegence is unifom ove compact sets. As a limiting fom of (.3) we obtain (x) (x) dv (x) (n 1) (x) dv (x) : Since jj = 1 almost eveywhee, we obtain as in the smooth case e (n 1)(x) (x) dv (x) 0: Fom now on, we assume lim i!1 v (g i ) = n 1. Lemma. We have v (g) = n 1.
PINCHING THEOREM FOR THE VOLUME ENTROPY 5 Poof. Since g i! g in C 1;, fo any " > 0 we have fo any v T f M fo i su ciently lage. It follows that B i (1 ") jvj gi jvj g (1 + ") jvj gi 1 + " B ( ) B i whee B i denotes a geodesic ball w..t. g i. Hence log volb i 1+" log volb ( ) Taking limit as! 1 yields v (g i ) 1 + " v (g) v (g i) 1 " : ; 1 " log volb i 1 " : As lim i!1 v (g i ) = n 1, we have v (g) = n 1. Fom (.), in view of Lemma 1 and Lemma, we can now conclude as in [LW] that fo o -a.e. @ c M e v(x) = 0 in the sense of distibution. By elliptic egulaity, C ;. We claim that D = g d d. To see this, st by the Bochne fomula we have fo any f C 1 and Cc 1 with 0 1 jfj g i i dv i h = D f i + hf; g i fi i gi + Ric gi (f; f) dv i hd f i g i + hf; i fi gi (n 1) jfj g i dv i h = D f i ( g i f) (n 1) jfj i g i hf; i gi i f dv i : By appoximation, we can take f to be in this fomula which in the limit, as i! 1, yields 0 = 1 jj dv h D () jj h; i i dv hd = i (n 1) (n 1) (n 1) h; i dv h = D i (n 1) (n 1) + (n 1) dv D = (n 1) dv; i.e. D n 1. On the othe hand, D (; ) = 1 = n 1. It is then obvious that D = g d d. D ; jj E = 0 and
6 FRANÇOIS LEDRAPPIER AND XIAODONG WANG It then follows that f M must be isometic to R with the metic g = dt + e t h ([LW, Theoem 6]). Suppose fu i g is a local chat on. If g is smooth, then a simple calculation shows e 4t R ijkl = e 4t R h ijkl (h ik h jl h il h jk ) : As g is only C 1;, this should be intepeted in the weak sense. As M f coves the compact M, the cuvatue on the left hand side is bounded unifomly in t. Theefoe h must be at in the weak sense. As a esult, g is Einstein in the weak sense. It is well known (cf. [P1]) that in a local hamonic coodiate system fx i g this leads to a elliptic system in the weak sense whee Q ij (g; @g) = 1 gkl @ g ij @x k @x l = (n 1) g ij + Q ij (g; @g) ; k ij l kl 1 @g kl @g il 1 @x j @x k l ik k jl + 1 @g kl @gil + @g jl @x k @x j @x i @g kl @x i @g jl @x k : By elliptic egulaity, the hamonic coodinates ae C ; w..t. the oiginal smooth stuctue on M. f But they ae smoothly compatible with one anothe and theefoe de ne a new smooth stuctue on M f w..t. which g is smooth. Moeove is also smooth and so is the decomposition M f = R with g = dt + e t h. Thenwe can conclude that h is at and (; h) is simply the at R n 1. Theefoe ; g is the hypebolic space H n. 3. Poof Theoem 6 and Futhe Remaks We st ecall the following facts on negatively cuved compact manifolds. Theoem 8. (Gomov [G1]) Suppose M n is a closed Riemannian manifold with 1 K M < 0. Then Thee exists C n > 0 s.t. vol (M) C n ; Fo n 8, thee exist c n > 0 s.t. vol (M) c n (1 + d (M)); Fo 4 n 7, thee exist c n > 0 s.t. vol (M) c n 1 + d 1=3 (M). We now pove Theoem 6. It su ces to pove that M is close to a hypebolic manifold in the Gomov-Hausdo sense. Suppose this is not tue, then we have a sequence (M n i ; g i) satisfying (1) g i has negative sectional cuvatue, () Ric (g i ) (n 1), (3) diam (M i ; g i ) D, (4) the volume entopy v (g i )! n 1: such that (M n i ; g i) is not close to any hypebolic manifold in Gomov-Hausdo sense. The sectional cuvatue of g i is bounded between (n 1) and 0. By Theoem 8, vol (M i ; g i ) C n > 0. Theefoe by the Cheege niteness theoem, we can assume
PINCHING THEOREM FOR THE VOLUME ENTROPY 7 that M i ae all di eomophic to a same manifold M passing to subsequence. By Gomov s convegence theoem [GLP, GW, P], we can nd di eomophisms f i such that the metics eg i = f i g i conveges to a metic g in C 1; passing to a subsequence. By Theoem 7, (M; g) is a hypebolic manifold. This is a contadiction. Remak 1. A natual question is if one can elax the negative cuvatue assumption in Theoem 6 by an abitay uppe bound on the sectional cuvatue. The di culty is to ule out collapsing. In geneal, collapsing can happen of couse. But it seems plausible that collapsing can be uled out unde the assumption Ric (g) (n 1) and v (g) is close to n 1. Refeences [A] E. Auby, Pincement su le specte et le volume en coubue de Ricci positive. Ann. Sci. École Nom. Sup. (4) 38 (005), no. 3, 387 405. [ChC1] J. Cheege and T. Colding, On the stuctue of spaces with Ricci cuvatue bounded below I, J. Di eential Geom. 46 (1997), 406 480. [ChC] J. Cheege and T. Colding, Lowe bounds on Ricci cuvatue and the almost igidity of waped poducts. Annals of Math. 144 (1996), 189-37. [ChC3] J. Cheege and T. Colding, On the stuctue of spaces with Ricci cuvatue bounded below. I. J. Di eential Geom. 46 (1997), no. 3, 406 480. [Co1] T. Colding, Shape of manifolds with positive Ricci cuvatue, Invent. Math. 14 (1996), 175 191. [Co] T. Colding, Lage manifolds with positive Ricci cuvatue, Invent. Math. 14 (1996), 193 14. [C] G. Coutois, Pincement spectal des vaiétés hypeboliques. Pepint, 000. [GW] R. Geene and H. Wu, Lipschitz convegence of Riemannian manifolds. Paci c J. Math. 131 (1988), no. 1, 119 141. [G1] M. Gomov, Manifolds of negative cuvatue. J. Di eential Geom. 13 (1978), no., 3 30. [GLP] M. Gomov, J. Lafontaine and P. Pansu, Stuctues métiques pou les vaiétés Riemanniennes. Cedic Nathan, Pais, 1981. [Kn] G. Kniepe, Spheical means on compact Riemannian manifolds of negative cuvatue. Di eential Geom. Appl. 4 (1994), no. 4, 361 390. [LW] F. Ledappie and X. Wang, An integal fomula fo the volume entopy with applications to igidity. Jounal Di. Geom. 85 (010), 461-478. [L] J. Lohkamp, Cuvatue h-pinciples. Ann. of Math. () 14 (1995), no. 3, 457 498. [P] S. Petes, Convegence of Riemannian manifolds, Compositio Math. 6 (1987), no. 1, 3-16. [P1] P. Petesen, Riemannian Geomety, second edition. GTM 171. Spinge-Velag, 006. [P] P. Petesen, On eigenvalue pinching in positive Ricci cuvatue. Invent. Math. 138 (1999), no. 1, 1 1. [P3] P. Petesen, On eigenvalue pinching in positive Ricci cuvatue, Eatum, Invent. Math. 155 (004) 3. [W] X. Wang, Hamonic functions, entopy, and a chaacteization of the hypebolic space. J. Geom. Anal. 18 (008), 7-84. LPMA, C.N.R.S. UMR7599, Boîte Couie 188-4, Place Jussieu 755 Pais cedex 05, Fance E-mail addess: fledapp@nd.edu Depatment of Mathematics, Michigan State Univesity, East Lansing, MI 4884, USA E-mail addess: xwang@math.msu.edu