Diastatic entropy and rigidity of hyperbolic manifolds
aa r X i v : . [ m a t h . DG ] M a y DIASTATIC ENTROPY AND RIGIDITY OF HYPERBOLICMANIFOLDS
ROBERTO MOSSA
Abstract.
Let f : Y → X be a continuous map between a compact realanalytic K¨ahler manifold ( Y, g ) and a compact complex hyperbolic manifold(
X, g ). In this paper we give a lower bound of the diastatic entropy of ( Y, g )in terms of the diastatic entropy of (
X, g ) and the degree of f . When thelower bound is attained we get geometric rigidity theorems for the diastaticentropy analogous to the ones obtained by G. Besson, G. Courtois and S.Gallot [2] for the volume entropy. As a corollary, when X = Y , we show thatthe minimal diastatic entropy is achieved if and only if g is isometric to thehyperbolic metric g . Contents
1. Introduction and statement of main results 12. Diastasis and diastasic entropy 33. Proof of Theorem 1.1 and Corollaries 1.1, 1.2 and 1.3 4References 81.
Introduction and statement of main results
In this paper, we define the diastatic entropy
Ent d ( Y, g ) of a compact real ana-lytic K¨ahler manifold (
Y, g ) with globally defined diastasis function (see Definition2.1 and 2.2 below). This is a real analytic invariant defined, in the noncompactcase, by the author in [17], where the link with Donaldson’s balanced condition isstudied. The diastatic entropy extends the concept of volume entropy using thediastasis function instead of the geodesic distance. Throughout this paper a com-pact complex hyperbolic manifold will be a compact real analytic complex manifold(
X, g ) endowed with locally Hermitian symmetric metric with holomorphic sec-tional curvature strictly negative (i.e. ( X, g ) is the compact quotient of a complexhyperbolic space, see Example 2.3 below). Our main result is the following theo-rem, analogous to the celebrated result of G. Besson, G. Courtois, S. Gallot on the Date : September 24, 2018. minimal volume entropy of a compact negatively curved locally symmetric manifold(see (12) below) [2, Th´eor`eme Principal]:
Theorem 1.1.
Let ( Y, g ) be a compact K¨ahler manifold of dimension n ≥ andlet ( X, g ) be a compact complex hyperbolic manifold of the same dimension. If f : Y → X is a nonzero degree continuous map, then Ent d ( Y, g ) n Vol (
Y, g ) ≥ | deg ( f ) | Ent d ( X, g ) n Vol (
X, g ) . (1) Moreover the equality is attained if and only if f is homotopic to a holomorphic oranti-holomorphic homothetic covering F : Y → X . As a first corollary we obtain a characterization of the hyperbolic metric as thatmetric which realizes the minimum of the diastatic entropy:
Corollary 1.1.
Let ( X, g ) be a compact complex hyperbolic manifold of dimension n ≥ and denote by E ( X, g ) the set of metrics g on X with globally defineddiastasis and fixed volume Vol ( g ) = Vol ( g ) . Then the functional F : E ( X, g ) → R ∪ {∞} given by g F Ent d ( X, g ) , attains its minimum when g is holomorphicallyor anti-holomorphically isometric to g . This corollary can be seen as the diastatic version of the A. Katok and M.Gromov conjecture on the minimal volume entropy of a locally symmetric spacewith strictly negative curvature (see [8, p. 58]), proved by G. Besson, G. Courtois,S. Gallot in [2]. We also apply Theorem 1.1 to give a simple proof for the complexversion of the Mostow and Corlette–Siu–Thurston rigidity theorems:
Corollary 1.2. (Mostow). Let ( X, g ) and ( Y, g ) be two compact complex hyper-bolic manifolds of dimension n ≥ . If X and Y are homotopically equivalent thenthey are holomorphically or anti-holomorphically homothetic. Corollary 1.3. (Corlette–Siu–Thurston). Let ( X, g ) and ( Y, g ) be as in the pre-vious corollary and with the same (constant) holomorphic sectional curvature. If f : Y → X is a continuous map such that Vol ( Y ) = | deg ( f ) | Vol ( X ) (2) then there exists a holomorphically or anti-holomorphically Riemannian covering F : Y → X homotopic to f . The paper consists of others two sections. In Section 2 we recall the basicdefinitions. Section 3 is dedicated to the proof of Theorem 1.1. The proof is basedon the analogous result for the volume entropy (see formula (12) below) and on F is said to be homothetic if F ∗ g = α g for some α > IASTATIC ENTROPY AND RIGIDITY OF HYPERBOLIC MANIFOLDS 3
Lemma 3.2 which provides a lower bound for the diastatic entropy in terms ofvolume entropy.
Acknowledgments.
The author would like to thank Professor Sylvestre Gallotand Professor Andrea Loi for their help and their valuable comments.2.
Diastasis and diastasic entropy
The diastasis is a special K¨ahler potential defined by E. Calabi in its seminalpaper [5]. Let (cid:16) e
Y , e g (cid:17) be a real analytic K¨ahler manifold. For every point p ∈ e Y there exists a real analytic function Φ : V → R , called K¨ahler potential, defined in aneighborhood V of p such that e ω = i ∂∂ Φ, where e ω is the K¨ahler form associated to e g . Let z = ( z , . . . , z n ) be a local coordinates system around p . By duplicating thevariables z and z the real analytic K¨ahler potential Φ can be complex analyticallycontinued to a function ˆΦ : U × U → C in a neighbourhood U × U ⊂ V × V of ( p, p )which is holomorphic in the first entry and antiholomorphic in the second one. Definition 2.1 (Calabi, [5]) . The diastasis function D : U × U → R is defined by D ( z, w ) := ˆΦ ( z, z ) + ˆΦ ( w, w ) − ˆΦ ( z, w ) − ˆΦ ( w, z ) . The diastasis function centered in w , is the K¨ahler potential D w : U → R around w given by D w ( z ) := D ( z, w ) . We will say that a compact K¨ahler manifold (
Y, g ) has globally defined diastasis ifits universal K¨ahler covering (cid:16) e
Y , e g (cid:17) has globally defined diastasis D : e Y × e Y → R .One can prove that the diastasis is uniquely determined by the K¨ahler metric e g and that it does not depend on the choice of the local coordinates system or on thechoice of the K¨ahler potential Φ.Calabi in [5] uses the diastasis to give necessary and sufficient conditions for theexistence of an holomorphic isometric immersion of a real analytic K¨ahler manifoldsinto a complex space form. For others interesting applications of the diastasisfunction see [10, 11, 12, 13, 14, 15, 18] and reference therein.Assume that (cid:16) e Y , e g (cid:17) has globally defined diastasis D : e Y × e Y → R . Its (normal-ized ) diastatic entropy is defined by:Ent d (cid:16) e Y , e g (cid:17) = X ( e g ) inf (cid:26) c ∈ R + : Z e Y e − c D w ν e g < ∞ (cid:27) , (3)where X ( e g ) = sup y, z ∈ e Y k grad y D z k and ν e g is the volume form associated to e g .If X ( e g ) = ∞ or the infimum in (3) is not achieved by any c ∈ R + , we set Our definition of diastatic entropy differs respect to the one given in [17] by the normalizingfactor X ( e g ). R. MOSSA
Ent d (cid:16) e Y , e g (cid:17) = ∞ . The definition does not depend on the base point w , indeed, as |D w ( x ) − D w ( x ) | = |D x ( w ) − D x ( w ) | ≤ X ( e g ) ρ ( w , w ) , we have e − c X ( e g ) ρ ( w , w ) Z e Y e − c D w ( x ) ν e g ≤ Z e Y e − c D w ( x ) ν e g ≤ e c X ( e g ) ρ ( w , w ) Z e Y e − c D w ( x ) ν e g , therefore R e Y e − c D w ( x ) ν e g < ∞ if and only if R e − c D w ( x ) ν e g < ∞ . Definition 2.2.
Let (
Y, g ) be a compact K¨ahler manifold with globally defineddiastasis. We define the diastatic entropy of (
Y, g ) asEnt d ( Y, g ) = Ent d (cid:16) e Y , e g (cid:17) , where (cid:16) e Y , e g (cid:17) is the universal K¨ahler covering of ( Y, g ). Example 2.3.
Let C H n = (cid:8) z ∈ C n : k z k = | z | + · · · + | z n | < (cid:9) be the uni-tary disc endowed with the hyperbolic metric e g h of constant holomorphic sectionalcurvature −
4. The associated K¨ahler form and the diastasis are respectively givenby e ω h = − i ∂ ¯ ∂ log (cid:0) − k z k (cid:1) . and D h ( w, z ) = − log (cid:0) − k z k (cid:1) (cid:0) − k w k (cid:1) | − zw ∗ | ! . (4)Denote by ω e = i ∂ ¯ ∂ k z k the restriction to C H n of the flat form of C n . One has Z C H n e − α D h ω nh n ! = Z C H n (cid:0) − | z | (cid:1) α − n − ω ne n ! < ∞ ⇔ α > n, and by a straightforward computation one sees that X ( e g h ) = 2. We conclude by(3) that Ent d ( C H n , e g h ) = 2 n. (5) Remark 2.4.
It should be interesting to compute X ( g B ), where g B is the Bergmanmetric of an homogeneous bounded domain. This combined with the results ob-tained in [17], will allow us to obtain the diastatic entropy of this domains.3. Proof of Theorem 1.1 and Corollaries 1.1, 1.2 and 1.3
We start by recalling the definition of volume entropy of a compact Riemannianmanifold (
M, g ). Let π : (cid:16) f M , e g (cid:17) → ( M, g ) its riemannian universal cover. Wedefine the volume entropy of (
M, g ) asEnt v ( M, g ) = inf (cid:26) c ∈ R + : Z f M e − c e ρ ( w, x ) ν e g ( x ) < ∞ (cid:27) , (6) IASTATIC ENTROPY AND RIGIDITY OF HYPERBOLIC MANIFOLDS 5 where e ρ is the geodesic distance on (cid:16) f M , e g (cid:17) and ν e g is the volume form associated to e g . By the triangular inequality, we can see that the definition does not depend onthe base point w . As the volume entropy depends only on the Riemannian universalcover it make sense to defineEnt v (cid:16) f M , e g (cid:17) = Ent v ( M, g ) . The classical definition of volume entropy of a compact riemannian manifold(
M, g ), is the followingEnt vol ( M, g ) = lim t →∞ t log Vol ( B p ( t )) , (7)where Vol ( B p ( t )) denotes the volume of the geodesic ball B p ( t ) ⊂ f M , of center in p and radius t . This notion of entropy is related with one of the main invariant forthe dynamics of the geodesic flow of ( M, g ): the topological entropy Ent top ( M, g )of this flow. For every compact manifold (
M, g ) A. Manning in [19] proved theinequality Ent vol ( M, g ) ≤ Ent top ( M, g ), which is an equality when the curvatureis negative. We refer the reader to the paper [2] (see also [3] and [4]) of G. Besson,G. Courtois and S. Gallot for an overview on the volume entropy and for the proofof the celebrated minimal entropy theorem. For an explicit computation of thevolume entropy Ent v (Ω , g ) of a symmetric bounded domain (Ω , g ) see [16].The next lemma shows that the classical definition of volume entropy (7) doesnot depend on the base point and it is equivalent to definition (6), that isEnt vol ( M, g ) = Ent v ( M, g ) . Lemma 3.1.
Denote by L := lim inf R → + ∞ (cid:18) R log (Vol B ( x , R )) (cid:19) and L := lim sup R → + ∞ (cid:18) R log (Vol B ( x , R )) (cid:19) , where B ( x , R ) ⊂ (cid:16) f M , e g (cid:17) is the geodesic ball of centre x and radius R . Then thetwo limits does not depends on x and L ≤ Ent v ( M, g ) ≤ L. Proof.
Let x an arbitrary point of M . Set D = d ( x , x ) and R > D . By thetriangular inequality B ( x , R − D ) ⊂ B ( x , R ) ⊂ B ( x , R + D ) . R. MOSSA
Let R ′ = R + D , we havelim inf R → + ∞ (cid:18) R log (Vol B ( x , R )) (cid:19) ≤ lim inf R → + ∞ (cid:18) R log (Vol B ( x , R + D )) (cid:19) = lim inf R ′ → + ∞ (cid:18) R ′ R ′ − D R ′ log (Vol B ( x , R ′ )) (cid:19) ≤ lim inf R ′ → + ∞ (cid:18) R ′ log (Vol B ( x , R ′ )) (cid:19) . With the same argument one can prove the inequality in the other direction, sothat L does not depend on x . Analogously we can prove that L does not dependon x .By the definition of limit inferior and superior, for every ε >
0, there exists R ( ε )such that, for R ≥ R ( ε ), L − ε ≤ (cid:18) R log (Vol B ( x , R )) (cid:19) ≤ L + ε equivalently e ( L − ε ) R ≤ (Vol B ( x , R )) ≤ e ( L + ε ) R . (8)Integrating by parts we obtain I := Z f M e − c e ρ ( x , x ) dv ( x ) = Z ∞ e − c r Vol n − ( S ( x , r )) dr = Vol ( B ( x , r )) e − c r (cid:12)(cid:12)(cid:12) ∞ + c Z ∞ e − c r Vol ( B ( x , r )) dr. where S ( x , r ) = ∂B ( x , r ). On the other hand, by (8) we get Z ∞ R ( ε ) e ( L − c − ε ) r dr ≤ Z ∞ R ( ε ) e − c r Vol ( B ( x , r )) dr ≤ Z ∞ R ( ε ) e − ( c − L − ε ) r dr. We deduce that if c > L then I is convergent i.e L ≥ Ent v and that if I is notconvergent when c < L , that is Ent v ≥ L , as wished. (cid:3) The next lemma show that the diastatic entropy is bounded from below by thevolume entropy.
Lemma 3.2.
Let ( Y, g ) be a compact K¨ahler manifold with globally defined dias-tasis, then Ent d ( Y, g ) ≥ Ent v ( Y, g ) . (9) This bound is sharp when ( Y, g ) is a compact quotient of the complex hyperbolicspace. That is, Ent d ( C H n , e g h ) = 2 n = Ent v ( C H n , e g h ) . (10) Proof.
Let ( e Y , e g ) be universal K¨ahler cover of ( Y, g ). For every w, x ∈ e Y we have D w ( x ) = D w ( x ) − D w ( w ) ≤ sup z ∈ e Y k d z D w k ρ w ( x ) ≤ X ( e g ) ρ w ( x ) , IASTATIC ENTROPY AND RIGIDITY OF HYPERBOLIC MANIFOLDS 7 so Z e Y e − c X ( e g ) ρ w ( x ) ν e g ≤ Z e Y e − c D w ( x ) ν e g . Therefore, if c X ( e g ) ≤ Ent v ( e Y , e g ) then c X ( e g ) ≤ Ent d ( e Y , e g ). We obtain (9) bysetting c = Ent v ( e Y , e g ) X ( e g ) . Equation (10) follow by (5) and [16, Theorem 1.1]. (cid:3) Proof of Theorem 1.1.
Let (
X, g ) as in Theorem 1.1 and let π X : ( C H n , e g ) → ( X, g ) be the universal covering. Notice that e g = λ e g h for some positive λ . Thenwe have Vol ( X, g ) Ent v ( X, g ) n = Vol ( X, g h ) Ent v ( X, g h ) n = Vol ( X, g h ) Ent d ( X, g h ) n = Vol ( X, g ) Ent d ( X, g ) n , (11)where the first and the third equality are consequence of the fact that Ent v ( C H n , e g ) = √ λ Ent v ( C H n , e g h ) and Ent d ( C H n , e g ) = √ λ Ent d ( C H n , e g h ), while the secondequality follows by (10). Let f : Y → X be as in Theorem 1.1, then, by [2,Th´eor`eme Principal] we know thatEnt v ( Y, g ) n Vol (
Y, g ) ≥ | deg ( f ) | Ent v ( X, g ) n Vol (
X, g ) (12)where the equality is attained if and only if f is homotopic to a homothetic covering F : Y → X . Putting together (9), (11) and (12) we get thatEnt d ( Y, g ) n Vol (
Y, g ) ≥ | deg ( f ) | Ent d ( X, g ) n Vol (
X, g )where the equality is attained if and only if f is homotopic to a homothetic covering F : Y → X .To conclude the proof it remains to prove that F is holomorphic or anti-holo-morphic. Up to homotheties, it is not restrictive to assume that g = F ∗ g , so thatits lift e F : e Y → C H n to the universal covering it is a global isometry. Fix a point q ∈ e Y , let p = e F ( q ) and denote A q = e F ∗ J p the endomorphism acting on T q e Y ,where J is the complex structure of C H n . Denote by G e Y and respectively G C H n theholonomy groups of ( e Y , e g ) and respectively ( C H n , e g ). Note that G e Y = e F ∗ G C H n and that G C H n = SU ( n ), therefore G e Y acts irreducibly on T q e Y . As J commuteswith the action of G C H n , by construction A q is invariant with respect to the action of G e Y . Therefore, denoted Id q the identity map of T q e Y , by Schur’s lemma, A q = λ Id q with λ ∈ C . Moreover − Id q = A q = λ Id q , so λ = ± i . By the arbitrarity of q weconclude that e F is holomorphic or anti-holomorphic. Proof of Corollary 1.1.
This is an immediate consequence of Theorem 1.1 onceassumed Y = X , Vol ( g ) = Vol ( g ) and f = id X the identity map of X . Proof of Corollary 1.2.
Let h : Y → X be an homotopic equivalence and h − itshomotopic inverse. Substituting in (1), once with f = h and once with f = h − , R. MOSSA we have respectivelyEnt d ( Y, g ) n Vol (
Y, g ) ≥ | deg ( h ) | Ent d ( X, g ) n Vol (
X, g )and Ent d ( X, g ) n Vol (
X, g ) ≥ (cid:12)(cid:12) deg (cid:0) h − (cid:1)(cid:12)(cid:12) Ent d ( Y, g ) n Vol (
Y, g ) . We then conclude that Ent d ( Y, g ) n Vol (
Y, g ) = Ent d ( X, g ) n Vol (
X, g ) andthat | deg ( h ) | = 1. Therefore, by applying the last part of Theorem 1.1, we see that h is homotopic to an holomorphic (or antiholomorphic) homothety F : X → Y . Proof of Corollary 1.3.
Let π Y : ( C H n , e g ) → ( Y, g ) and π X : ( C H n , e g ) → ( X, g ) be the universal coverings, since g and g are both hyperbolic with thesame curvature, we conclude that e g = e g and that Ent d ( X, g ) = Ent d ( Y, g ).Therefore we get an equality in (1). Using again the last part of Theorem 1.1 weget Vol ( Y ) = | deg ( F ) | Vol ( X ) and we conclude that F is locally isometric. References [1]
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