The CAT(0) geometry of convex domains with the Kobayashi metrics
aa r X i v : . [ m a t h . C V ] N ov THE CAT(0) GEOMETRY OF CONVEX DOMAINS WITH THEKOBAYASHI METRICS
JINSONG LIU & HONGYU WANG
Abstract.
Let (Ω , K Ω ) be a convex domain in C d with the Kobayashi metric K Ω . In this paper we prove that m -convexity is a necessary condition for(Ω , K Ω ) to be CAT(0) if d = 2. Moreover, when Ω ⊂ C d , d ≥
3, we obtain asimilar result with the further smoothness assumption on its boundary. Introduction
A CAT(0) space is a geodesic metric space whose geodesic triangles are slimmerthan the corresponding flat triangles in the Euclidean plane R . CAT(0) spaces arenatural generalizations of complete simply connected manifolds with nonpositivesectional curvature. Refer to [3] for more details.Our main result is the following. Theorem 1.1. (1) If Ω ⊂ C is a C -proper convex domain and (Ω , K Ω ) isCAT(0), then Ω is locally m-convex for some m ∈ N .(2) Suppose that Ω ⊂ C d ( d ≥ is a bounded convex domain with smoothboundary. If (Ω , K Ω ) is CAT(0), then ∂ Ω has finite line type. Recall that a convex domain is called C -proper if Ω does not contain any entirecomplex affine lines. There is a well-known result on C -proper convex domains. Proposition 1.2 ( [2]) . If Ω is a C -proper convex domain in C d , then the Kobayashimetric K Ω is complete. A C -proper convex domain is called locally m-convex if for any R >
C > m ≥ z ∈ B (0 , R ) ∩ Ω and non-zero v ∈ C d , δ Ω ( z, v ) ≤ Cδ m Ω ( z ) . Note that m -convexity is related to finite type by the following proposition. Proposition 1.3 ( [8], Proposition 9.1) . Given a bounded convex domain Ω withsmooth boundary, then Ω is m -convex for some m ∈ N if and only if ∂ Ω has finiteline type in the sense of D’Angelo. The first author is supported by NSF of China No.11671057 and No.11688101.
Motivation from Gromov Hyperbolicity.Definition 1.4.
Let (
X, d ) be a metric space. Given three points x, y, o ∈ X, theGromov product is given by( x | y ) o = 12 ( d ( x, o ) + d ( o, y ) − d ( x, y )) . A proper geodesic metric space (
X, d ) is Gromov hyperbolic if and only if thereexists δ ≥ o, x, y, z ∈ X ,( x | y ) o ≥ min { ( x | z ) o , ( z | y ) o } − δ. Z M. Balogh and M. Bonk [1] firstly proved those strongly pseudoconvex domainsequipped with the Kobayashi metric are Gromov hyperbolic. Later A M. Zimmer[8] proved that smooth convex domains with the Kobayashi metrics are Gromovhyperbolic if and only if they are of finite type.Recently Zimmer proved that locally m -convexity is a necessary condition forthose convex domains to be Gromov hyperbolicity. Theorem 1.5 ( [7], Corollary 7.2) . Suppose that Ω is a C -proper convex domainand (Ω , K Ω ) is Gromov hyperbolicity. Then Ω is locally m -convex. This paper is motivated by the above Zimmer’s work, and Theorem 1 . Preliminaries
Notations. (1) For z ∈ C d , let | z | be the standard Euclidean norm and let d euc ( z , z ) = | z − z | be the standard Euclidean distance.(2) Given an open set Ω ⊂ C n , p ∈ Ω and v ∈ C n \{ } , let δ Ω ( p ) = inf { d euc ( p, x ) : x ∈ ∂ Ω } as before, and let δ Ω ( p, v ) = inf { d euc ( p, x ) : x ∈ ∂ Ω ∩ ( p + C v ) } . (3) For any curve σ , we denote by L ( σ ) the length of σ .(4) For any z ∈ C n and δ >
0, let B euc ( z , δ ) denote the open ball B euc ( z , δ ) = { z ∈ C n | | z − z | < δ } .(5) Write Af f ( C d ) the group of complex affine automorphisms of C d .(6) Let X d denote the set of all C -proper convex domains in C d endowed withthe local Hausdorff topology.2.2. The Kobayashi metric.
Given a domain Ω ⊂ C d , the (infinitesimal) Kobayashimetric is the pseudo-Finsler metric defined by k Ω ( x ; v ) = inf {| ξ | : f ∈ Hol( D , Ω) , f (0) = x, d ( f ) ∗ , ( ξ ) = v } . Define the length of any curve σ to be L ( σ ) = Z ba k Ω ( σ ( t ); σ ′ ( t )) dt. HE CAT(0) GEOMETRY OF CONVEX DOMAINS WITH THE KOBAYASHI METRICS 3
Then we can define the Kobayashi pseudo-distance to be K Ω ( x, y ) = inf { L ( σ ) | σ : [ a, b ] → Ω is any absolutely continuous curvewith σ ( a ) = x and σ ( b ) = y } . The following is a well known property on the Kobayashi metric.
Proposition 2.1. If f : Ω → Ω is holomorphic, then, for all z ∈ Ω and v ∈ C d , k Ω ( f ( z ); df z ( v )) ≤ k Ω ( z ; v ) . Moreover, K Ω ( f ( z ) , f ( z )) ≤ K Ω ( z , z ) , for all z , z ∈ Ω . For any product domain, the Kobayashi metric has the following product prop-erty (cf. [6], p.107), K Ω × Ω (( x , y ) , ( x , y )) = max { K Ω ( x , x ) , K Ω ( y , y ) } , which makes a product domain behave like a positively curved space.2.3. CAT(0) space.Definition 2.2.
Let I ⊂ R be an interval. A map σ : I → Ω is called a geodesicsegment if, for all s, t ∈ I , K Ω ( σ ( s ) , σ ( t )) = | t − s | . And (
X, d ) is called a geodesic metric space if any two points in X are joined by ageodesic segment. Remark . Note that the paths which are commonly called ’geodesics’ in differ-ential geometry need not be geodesics in the above sense. In general they will onlybe local geodesics.Let (
X, d ) be a geodesic metric space. For any three points a, b, c ∈ X , supposethat [ a, b ] , [ b, c ] , [ c, a ] form a geodesic triangle ∆. Let ¯∆(¯ a, ¯ b, ¯ c ) ⊂ R be a triangle inthe Euclidean plane with the same edge lengths as ∆. Let p, q be any points on [ a, b ]and [ a, c ] , and let ¯ p, ¯ q be the corresponding points on [¯ a, ¯ b ] and [¯ a, ¯ c ] , respectively,such that dist X ( a, p ) = dist R (¯ a, ¯ p ) , dist X ( a, q ) = dist R (¯ a, ¯ q ) . Definition 2.4.
We call (
X, d ) a
CAT ( ) space, if for any geodesic triangle ∆ ⊂ X the inequality dist X ( p, q ) ≤ dist R (¯ p, ¯ q ) holds.Typical examples are trees and complete simply connected manifolds with non-positive sectional curvature. Note that there is an equivalent definition about CAT (0) spaces.
Theorem 2.5 ( [4]) . Let ( X, d ) be a geodesic metric space. Then ( X, d ) is CAT (0) if and only if for any three points x, y, z ∈ X , d ( z, m ) ≤
12 ( d ( z, x ) + d ( z, y )) − d ( x, y ) , where m is the midpoint of the geodesic segment from x to y . JINSONG LIU & HONGYU WANG
Finite type.
For any function f : C → R with f (0) = 0, we will denote by ν ( f ) the order of vanishing of f at 0.Let Ω = (cid:8) z ∈ C d : r ( z ) < (cid:9) where r is a C ∞ defining function with ∇ r = 0near ∂ Ω. A point x ∈ ∂ Ω is said to have finite line type L ifsup (cid:8) ν ( r ◦ ℓ ) | ℓ : C → C d is a non-trivial affine map and ℓ (0) = x (cid:9) = L Note that ν ( r ◦ ℓ ) ≥ ℓ ( C ) is tangent to Ω . Local Hausdorff topology.
Given a set A ⊂ C d , let N ǫ denote the ǫ -neighborhood of A with respect to the Euclidean distance. The Hausdorff distancebetween any two compact sets A , B is given by d H ( A, B ) = inf { ǫ > A ⊂ N ǫ ( B ) and B ⊂ N ǫ ( A ) } . The Hausdorff distance is a complete metric on the space of compact sets in C d .The space of all closed convex sets in C d can be given a topology from the localHausdorff semi-norms.For R > A ⊂ C d , let A ( R ) := A ∩ B R (0). Then we can define thelocal Hausdorff semi-norms by d ( R ) H ( A, B ) := d H ( A ( R ) , B ( R ) ) . Since an open convex set is completely determined by its closure, we say a sequenceof open convex sets { A n } converges in the local Hausdorff topology to an openconvex set B if d ( R ) H ( A, B ) → R >
Theorem 2.6 ( [8],Theorem 4.1) . Suppose that { Ω n } is a sequence of C -properconvex domains converging to a C -proper convex domain Ω in the local Hausdorfftopology. Then, for all x, y ∈ Ω , K Ω ( x, y ) = lim n →∞ K Ω n ( x, y ) , uniformly on compact sets of Ω × Ω . Proof of Theorem 1.1
Our proof is based on the following simple observation.
Observation 3.1. If Ω = Ω × Ω is a Kobayashi hyperbolic domain, then (Ω , K Ω ) is not a CAT (0) space.Proof.
Take x = y ∈ Ω and let m be the midpoint of the geodesic segment from x to y in (Ω , K Ω ). Then, we can choose z, w ∈ Ω such that K Ω ( z, w ) = K Ω ( x, m ) = K Ω ( y, m ) = 12 K Ω ( x, y ) , which implies that12 ( K (( x, w ) , ( m, z )) + K (( y, w ) , ( m, z ))) − K (( x, w ) , ( y, w )) = 0 . Since K Ω (( m, w ) , ( m, z )) >
0, it follows from Theorem 2.5 that (Ω , K Ω ) is not CAT (0). It completes the proof. (cid:3)
To prove Theorem 1 .
1, we shall need a recent result due to A M. Zimmer.
HE CAT(0) GEOMETRY OF CONVEX DOMAINS WITH THE KOBAYASHI METRICS 5
Theorem 3.2 ( [7], Theorem 6.1) . Suppose that Ω is a C -proper convex domain andevery domain in Af f ( C d ) · Ω T X d does not contain any affine disk in the boundary.Then Ω is locally m-convex for some m ≥ . The above theorem shows that: if Ω is not m-convex, then by scaling we canfind an affine disk in the boundary. By using the above theorem, the next Lemmais obvious.
Lemma 3.3.
Let Ω be a C -proper convex domain. If Ω is not locally m -convexfor any m ∈ N , then there exists A n ∈ Af f ( C d ) such that A n Ω → b Ω and b Ω ⊇ C ( α, β ) × ∆ × { ~ } , where C ( α, β ) = { z ∈ C : arg z ∈ ( α, β ) } is a convex cone.Proof. In view of Theorem 3 .
2, we may assume that { } × ∆ × { ~ } ⊂ ∂ Ω, andΩ ⊂ { ( z , ..., z d ) : Imz > } . Writing A n ( z ) = (cid:18) n I d − (cid:19) , we obtain A n (Ω ∩ C × { ~ } ) = C ( α, β ) × { ~ } , where C ( α, β ) = S t> t (Ω ∩ C × { ~ } ). (cid:3) Lemma 3.4.
Let Ω ⊂ C d be a C -proper convex domain. Suppose that P : C d → C is the projection map P ( z , ...z d ) = z . If Ω ∩ ( C × { ~ } ) = U × { ~ } and P (Ω) = U ,then the map F : U → Ω given by F ( z ) = ( z,~ induces an isometric embeddding ( U, K U ) → (Ω , K Ω ) .Proof. Since both F and P are holomorphic maps, from the distance decreasingproperty of the Kobayashi metrics, it follows that K Ω ( F ( z ) , F ( z )) ≤ K U ( z , z ) . Noting that P ◦ F = id , we thus have K U ( z , z ) ≤ K Ω ( F ( z ) , F ( z )) . (cid:3) Now we are ready to prove Theorem 1 . Proof.
Part (1) . We shall first prove the theorem when Ω ⊂ C is a C -properconvex domain.Assume, by contradiction, that Ω is not locally m -convex. From Lemma 3 . A n ∈ Af f ( C d ) such that Ω n := A n Ω → b Ω and b Ω ⊇ C ( α, β ) × ∆ and b Ω ∩ C = C ( α, β ).We claim that P ( b Ω) = C ( α, β ), where P ( z , z ) = z is the projection map.Suppose that it is not the case. Take p = ( z, ω ) ∈ b Ω, where z is not contained in C ( α, β ) and ω = | ω | e iθ . And take q = ( ξ, − e iθ ) where Im ξ = Imz and ξ lies inthe boundary of C ( α, β ) such that Reξ · Rez >
0. Since b Ω is also convex, it impliesthat { tp + (1 − t ) q : t ∈ (0 , } ⊂ b Ω . By taking t = | ω | +1 , we obtain that tz + (1 − t ) ξ ∈ b Ω, which contradicts with thefact that b Ω ∩ C = C ( α, β ). JINSONG LIU & HONGYU WANG
Then, by using Lemma 3 .
4, it follows that the map f : C ( α, β ) → b Ω given by f ( z ) = ( z,
0) induces an isometric embeddding( C ( α, β ) , K C ( α,β ) ) → ( b Ω , K b Ω ) . Now we choose x, y ∈ C ( α, β ) and let m be the midpoint of the geodesic segmentfrom x to y in the metric space ( C ( α, β ) , K C ( α,β ) ). Since f is isometric, m is also themidpoint of the geodesic segment from x to y in metric space ( b Ω , K b Ω ). Therefore,we can take z ∈ ∆ such that K ∆ (0 , z ) = K C ( α,β ) ( x, m ) = K C ( α,β ) ( m, y ) = 12 K C ( α,β ) ( x, y ) . Denote C = C ( α, β ) × ∆, ˆ x = ( x, y = ( y, m = ( m,
0) and ˆ z = (0 , z ).Since b Ω ⊇ C , it follows that K C (ˆ x, ˆ m ) ≥ K b Ω (ˆ x, ˆ m ) , and K C (ˆ y, ˆ m ) ≥ K b Ω (ˆ y, ˆ m ) . Therefore, 12 ( K b Ω (ˆ x, ˆ m ) + K b Ω (ˆ y, ˆ m )) − K b Ω (ˆ x, ˆ y ) ≤
12 ( K C (ˆ x, ˆ z ) + K C (ˆ y, ˆ z )) − K C (ˆ x, ˆ y )=0 . Choose x n , y n , z n ∈ Ω such that A n x n → ˆ x , A n y n → ˆ y and A n z n → ˆ z . NowTheorem 2 . K b Ω (ˆ x, ˆ y ) = lim n →∞ K Ω n ( A n x n , A n y n ) , and K b Ω (ˆ x, ˆ z ) = lim n →∞ K Ω n ( A n x n , A n z n ) , and K b Ω (ˆ y, ˆ z ) = lim n →∞ K Ω n ( A n y n , A n z n ) . Let m n be the midpoint of the geodesic segment from A n x n to A n y n in (Ω n , K Ω n ).Then, by choosing a subsequence (still denoted by m n ), we may suppose that m n → ˆ m ∈ b Ω ∪ {∞} . Then either ˆ m = ˆ z or ˆ m = ˇ z , where ˇ z = ( m, iz ).Since K ∆ (0 , z ) = K ∆ (0 , iz ), the equalities K C (ˆ x, ˆ z ) = K (ˆ x, ˇ z ) and K C (ˆ y, ˆ z ) = K C (ˆ y, ˇ z ) follow. We have thus proved that12 ( K b Ω (ˆ x, ˇ z ) + K b Ω (ˆ y, ˇ z )) − K b Ω (ˆ x, ˆ y ) ≤
12 ( K C (ˆ x, ˇ z ) + K C (ˆ y, ˇ z )) − K C (ˆ x, ˆ y )= 0 . Therefore, we deduce that: ∀ ǫ >
0, there exists N ∈ N such that ∀ n > N
12 ( K n (ˆ x, ˆ z ) + K n (ˆ y, ˆ z )) − K n (ˆ x, ˆ y ) ≤ ǫ, and 12 ( K n (ˆ x, ˇ z ) + K n (ˆ y, ˇ z )) − K n (ˆ x, ˆ y ) ≤ ǫ. HE CAT(0) GEOMETRY OF CONVEX DOMAINS WITH THE KOBAYASHI METRICS 7
Combining with the fact that ˆ m = ˆ z or ˆ m = ˇ z , we have thus proved that thereexists δ > K Ω n ( m n , ˆ z ) and K Ω n ( m n , ˇ z ) is strictly bigger than δ .Therefore, in terms of the definition of CAT (0) spaces, by choosing ǫ small enough,we complete the proof.Part (2) . Next we prove the result for the general case that Ω ⊂ C d , d ≥ d ≥
3, the claim P ( b Ω) = C ( α, β ) may be not correct without the further smoothnessassumption on the boundary.We will use the proof of Proposition 6.1 in [8]. For the sake of completeness, wepresent its proof here.Suppose ~ ∈ ∂ Ω andΩ ∩ O = { ~z ∈ O : Im ( z ) > f (Re ( z ) , z , . . . , z d ) } , where O is a neighborhood of the origin and f : R × C d − → R is a smooth convexnon-negative function. Assuming that ~ z → f (0 , z, , . . . , | z | n = 0 . Then there are two cases (a) (b):(a). If ∂ Ω contains an affine disk at ~
0, without losing of generality we assume that ~ × ∆ × { ~ } ⊂ ∂ Ω. By taking A n ( z ) = (cid:18) n I d − (cid:19) , we deduce that A n (Ω) → b Ω, and H × ∆ × { ~ } ⊂ b Ω , where H is the upper half plane.Since b Ω ⊂ { z ∈ C d : Imz > } , by considering the projection P : C d → C , P ( z , ..., z d ) = z , we obtain P ( b Ω) = H . Therefore, the map f : H → b Ω given by f ( z ) = ( z,~
0) induces an isometric embed-dding ( H , K H ) → ( b Ω , K b Ω ).Then by repeated use of the proof of Part (1), we deduce that Ω is not CAT(0).(b). Assume that ∂ Ω does not contain any affine disks at { ~ } . Similarly we onlyneed to check that H × ∆ × { ~ } ⊂ b Ω.The proof of the theorem could be simplified if we use the following lemma.
Lemma 3.5 ( [5], Theorem 9.3) . Suppose Ω ⊂ C d is a C -proper convex openset. Suppose that V ⊂ C d is a complex affine subspace intersecting Ω and { A n ∈ Af f ( V ) } is a sequence of affine maps such that A n (Ω ∩ V ) converges in the localHausdorff topology to a C -proper convex open set b Ω V ⊂ V . Then there exists affinemaps B n ∈ Aff (cid:0) C d (cid:1) such that B n Ω converges in the local Hausdorff topology to a C -proper convex open set b Ω with b Ω ∩ V = b Ω V . JINSONG LIU & HONGYU WANG
Now suppose V = C × { ~ } and Ω V = Ω ∩ V . Let G, W ⊂ R and U ⊂ C beneighborhoods of 0 such that f : G × U → W andΩ V ∩ O = { ( x + iy, z ) : x ∈ G, z ∈ U, y > f ( x, z ) } , where O = ( G + iW ) × U . By rescaling we may assume that B (0) ⊂ U . We canfind a n → z n ∈ B (0) such that f (0 , z n ) = a n | z n | n and f (0 , w ) ≤ a n | w | n forall w ∈ C with | w | ≤ | z n | .By the hypothesis that ∂ Ω V has no non-trivial complex affine disks, we obtainthat z n → f (0 , z n ) →
0. Passing to a subsequence we may assumethat | f (0 , z n ) | < . Consider the sequence of linear transformations A n = (cid:18) f (0 ,z n ) z − n (cid:19) , n = 1 , , · · · , and Ω V n = A n Ω V → b Ω V . Therefore,Ω V n ∩ O n = { ( x + iy, z ) : x ∈ G n , z ∈ U n , y > f n ( x, z ) } , where G n = f (0 , z n ) − G, U n = z − n U , and O n = A n O , and f n ( x, z ) = 1 f (0 , z n ) f ( f (0 , z n ) x, z n z ) . For | w | <
1, we then have f n (0 , w ) = f (0 , z n w ) f (0 , z n ) ≤ a n | z n | n | w | n f (0 , z n ) = | w | n , which implies that { } × ∆ ⊂ ∂ b Ω V . By using Ω V n ∩ ( C × { } ) = 1 f (0 , z n ) (Ω V ∩ ( C × { } )) , and f (0 , z n ) →
0, we have H × { } ⊂ b Ω V . Since b Ω V is convex, H × ∆ ⊂ b Ω V is valid.It follows immediately from Lemma 3 . B n ∈ Af f ( C d ) such that B n Ω → b Ω and H × ∆ × { ~ } ⊂ b Ω, which completes the proof. (cid:3)
It’s natural to ask whether the m -convexity is a sufficient condition for boundedconvex domains being CAT(0). However, the following example shows that m -convexity does not imply CAT(0) in general. Example 3.6 ( [7], Example 7.3) . Let Ω , Ω be bounded strongly convex domainsin C with C ∞ boundary. Furthermore, we assume ∈ ∂ Ω j , and the real hyperplane (cid:8) ( z , z ) ∈ C : Re ( z j ) = 0 (cid:9) is tangent to Ω j at , and Ω j ⊂ (cid:8) ( z , z ) ∈ C : Re ( z j ) > (cid:9) . Define
Ω = Ω ∩ Ω . HE CAT(0) GEOMETRY OF CONVEX DOMAINS WITH THE KOBAYASHI METRICS 9
Since each Ω j has smooth boundary, we see that ( ǫ, ǫ ) ∈ Ω for ǫ > j is strongly convex, it followsthat with a constant C > δ Ω j ( z ; v ) ≤ Cδ Ω j ( z ) / for all 1 ≤ j ≤ , z ∈ Ω j , and non-zero v ∈ C . Then, for z ∈ Ω and non-zero v ∈ C , we have δ Ω ( z ; v ) = min ≤ j ≤ δ Ω j ( z ; v ) ≤ min ≤ j ≤ Cδ Ω j ( z ) / = Cδ Ω ( z ) / , from which we deduce that Ω is 2-convex. However the set of domains { n · Ω } converges in the local Hausdorff topology to the domain D = (cid:8) ( z , z ) ∈ C : Re ( z ) > , Re ( z ) > (cid:9) . Thus ∂D contains an affine disk.Then, by repeated use of the proof of Theorem 1 .
1, we obtain that (Ω , K Ω ) isnot CAT(0). References [1] Z. M. Balogh and M. Bonk. Gromov hyperbolicity and the kobayashi metric on strictly pseu-doconvex domains.
Commentarii Mathematici Helvetici , 75(3):504–533, 2000.[2] Theodore J. Barth. Convex domains and kobayashi hyperbolicity.
Proceedings of the AmericanMathematical Society , 79(4):556–558, 1980.[3] Haefliger A. Bridson M R.
Metric spaces of non-positive curvature . 2013.[4] Fran¸cois Bruhat and Jacques Tits. Groupes r´eductifs sur un corps local: I. donn´ees radiciellesvalu´ees.
Publications Math´ematiques de l’IH ´ES , 41:5–251, 1972.[5] Sidney Frankel. Applications of affine geometry to geometric function theory in several complexvariables. part i. convergent rescalings and intrinsic quasiisometric structure.
Several ComplexVariables and Complex Geometry (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math , 52:183–208, 1991.[6] Marek Jarnicki and Peter Pflug.
Invariant distances and metrics in complex analysis-revisited .1993.[7] Andrew Zimmer. Subelliptic estimates from gromov hyperbolicity.[8] Andrew M. Zimmer. Gromov hyperbolicity and the kobayashi metric on convex domains offinite type.
Mathematische Annalen , 365(3):1–74, 2016.
HLM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences,Beijing, 100190, China & School of Mathematical Sciences, University of Chinese Acad-emy of Sciences, Beijing, 100049, China
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