Kobayashi Metric Research Articles

Sharp estimates of the Kobayashi metric and Gromov hyperbolicity

Let D = { ρ < 0 } be a smooth relatively compact domain in a four-dimensional almost complex manifold ( M , J ) , where ρ is a J-plurisubharmonic function on a neighborhood of D ¯ and strictly J-plurisubharmonic on a neighborhood of ∂ D. We give sharp estimates of the Kobayashi metric. Our approach is based on an asymptotic quantitative description of both the domain D and the almost complex structure J near a boundary point. Following Z.M. Balogh and M. Bonk [Z.M. Balogh, M. Bonk, Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comment. Math. Helv. 75 (2000) 504–533], these sharp estimates provide the Gromov hyperbolicity of the domain D.

ON THE SINGULAR KOBAYASHI PSEUDOMETRICS

In this paper, we'll introduce infinitesimal Finsler metrics induced by the higher order Kobayashi pseudometric and then prove some properties related to these as compared with the usual Kobayashi metric.

On the dynamics of isometries

We provide an analysis of the dynamics of isometries and semicontractions of metric spaces. Certain subsets of the boundary at infinity play a fundamental role and are identified completely for the standard boundaries of CAT(0)-spaces, Gromov hyperbolic spaces, Hilbert geometries, certain pseudoconvex domains, and partially for Thurston's boundary of Teichmueller spaces. We present several rather general results concerning groups of isometries, as well as the proof of other more specific new theorems, for example concerning the existence of free nonabelian subgroups in CAT(0)-geometry, iteration of holomorphic maps, a metric Furstenberg lemma, random walks on groups, noncompactness of automorphism groups of convex cones, and boundary behaviour of Kobayashi's metric.

The structure equations of a complex Finsler manifold

For a strongly pseudo-convex complex Finsler manifold M, a bundle U of adapted unitary frames is canonically defined. A non-linear Hermitian connection on U, invariant under local biholomorphic isometries, is given and it proved to be unique. By means of such connection, an absolute parallelism on U is determined and a new set of structure functions which generate all the isometric invariants of a Finsler metric is obtained. A pseudo-convex complex Finsler manifolds M, which admits a totally geodesic complex curve with a given constant holomorphic sectional curvature through any point and any direction, is called E-manifold. Main examples of E-manifolds are the smoothly bounded, strictly convex domains in C^n, endowed with the Kobayashi metric. A complete characterization of E-manifolds, using the previously defined structure functions, is given and a smaller set of generating functions for the isometric invariants of E-manifolds is determined.

Holomorphic maps into complex ellipsoids which are kobayashi isometries at one point

Let M be a connected taut complex manifold of dimension n and let D be a bounded balanced pseudoconvex domain in with continuous Minkowski function. Assume that there exist a finite number of complex hyperplanes H j through the origin such that every point of is an extreme point for . Let f:→D be a holomorphic map. Let p be a point of M. Assume that f(p) = 0 and that df p is an isometry for the infinitesimal Kobayashi metric. In this case, we will show that f is a biholomorphic map.

Holomorphic mappings into bounded complete reinhardt domains of holomorphy which are kobayashi isometries at one point

Let M be a connected taut complex manifold of dimension n such that the set H(M) of holomorphic functions on M separates points of M and let D be a bounded complete Reinhardt domain of holomorphy in Let f:M→D be a holomorphic mapping. Let p be a point of M. Assume that f(p)=0 and that df p is an isometry for the infinitesimal Kobayashi metric, in this case, we will show that f is a biholomorphic mapping

Holomorphic curvature of Finsler metrics and complex geodesics

In his famous 1981 paper, Lempert proved that given a point in a strongly convex domain the complex geodesics (i.e., the extremal disks) for the Kobayashi metric passing through that point provide a very useful fibration of the domain. In this paper we address the question whether, given a smooth complex Finsler metric on a complex manifoldM, it is possible to find purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesies ofM. We first discuss at some length the notion of holomorphic sectional curvature for a complex Finsler metric; then, using the differential equation of complex geodesies we obtained in [AP], we show that for every pair (p;v) ∈T M, withv ≠ 0, there is a (only a segment if the metric is not complete) complex geodesic passing throughp tangent tov iff the Finsler metric is Kähler, has constant holomorphic sectional curvature −4, and its curvature tensor satisfies a specific simmetry condition—which are the differential geometric conditions we were after. Finally, we show that a complex Finsler metric of constant holomorphic sectional curvature −4 satisfying the given simmetry condition on the curvature is necessarily the Kobayashi metric.

On the analytic structure of certain infinite dimensional Teicmüller spaces

It is well known since long time that quasiconformally different finite Riemann surfaces give rise to biholomorphically nonequivalent Teichmüller spaces except for a few obvious cases (cf. [R], [E-K]). This is deduced as an application of Royden’s theorem asserting that the Teichmüller metric is equal to the Kobayashi metric. For the case of infinite Riemann surfaces, however, it is still unknown whether or not the corresponding result holds, although it has been shown by F. Gardiner [G] that Royden’s theorem is also valid for the infinite dimensional Teichmüller spaces. On the other hand, recent activity of several mathematicians shows that the infinite dimensional Teichmüller spaces are interesting objects of complex analytic geometry (cf. [Kru], [T], [N], [E-K-K]). Therefore, based on the generalized form of Royden’s theorem, one might well look for further insight into Teichmüller spaces by studying the above mentioned nonequivalence question.

Asymptotic Expansions of Invariant Metrics of Strictly Pseudoconvex Domains

AbstractIn this paper we obtain the asymptotic expansions of the Carathéodory and Kobayashi metrics of strictly pseudoconvex domains with C∞ smooth boundaries in ℂn. The main result of this paper can be stated as following:Main Theorem. Let Ω be a strictly pseudoconvex domain with C∞ smooth boundary. Let FΩ(z,X) be either the Carathéodory or the Kobayashi metric of Ω. Let δ(z) be the signed distance from z to ∂Ω with δ(z) &lt; 0 for z ∊ Ω and δ(z) ≥ 0 for z ∉ Ω. Then there exist a neighborhood U of ∂Ω, a constant C &gt; 0, and a continuous function C(z,X):(U ∩ Ω) × ℂn -&gt; ℝ such that and|C(z,X)| ≤ C|X| for z ∊ U ∩ Ω and X ∊ ℂn

Smoothness of kobayashi metric of ellipsoids

Lempert proved that the Kobayashi metric and the Caratheodory metric of smooth strongly convex domains are smooth away from the zero section of the holomorphic tangent bundle. We will show that if the strong convexity is replaced by the weaker condition of strict convexity the above smoothness result is no longer valid even if the boundary is real analytic. We study the smoothness of the Kobayashi metric of ellipsoids. We prove that the Kobayashi metric of domains of the form , where m≥3/2, is piecewise C 3 off the zero section, but not C 3.

Some Estimates of the Kobayashi Metric in the Normal Direction

Some Estimates of the Kobayashi Metric in the Normal Direction

The limiting behavior of the Kobayashi-Royden pseudometric

We study the limit of the sequence of Kobayashi metrics of Riemann surfaces (when these Riemann surfaces form an analytic fibration in such a way that the total space of fibration becomes a complex surface), as the fibers approach the center fiber which is not in general smooth. We prove that if the total space is a Stein surface and the smooth part of the center fiber contains a component biholomorphic to a quotient of the disk by a Fuchsian group of first kind, then the Kobayashi metrics of the near-by fibers converge to the Kobayashi metric of this component as fibers tend to the center fiber.

BMO, VMO and Hankel operators on the Bergman space of strongly pseudoconvex domains

For bounded strongly pseudoconvex domains D with smooth boundary in C n , we introduce a kind of “mean oscillation” in terms of the Kobayashi metric. For f ϵ L 2( D), it is shown that if f has “bounded mean oscillation on D,” then the Hankel operators H f and H f from the Bergman space H 2( D), consisting of all holomorphic L 2 functions, into L 2( D) are bounded; if f has “vanishing mean oscillation at the boundary of D,” then H f and H f are compact. For f ϵ H 2( D), the conditions are also necessary.

The Boundary Behavior of the Kobayashi Metric

The Boundary Behavior of the Kobayashi Metric

Finsler metrics with properties of the Kobayashi metric on convex domains

The structure of complex Finsler manifolds is studied when the Finsler metric has the property of the Kobayashi metric on convex domains: (real) geodesics locally extend to complex curves (extremal disks). It is shown that this property of the Finsler metric induces a complex foliation of the cotangent space closely related to geodesics. Each geodesic of the metric is then shown to have a unique extension to a maximal totally geodesic complex curve S which has properties of extremal disks. Under the additional conditions that the metric is complete and the holomorphic sectional curvature is -4, S coincides with an extremal disk and a theorem of Faran is recovered: the Finsler metric coincides with the Kobayashi metric.

The Kobayashi and Bergman metrics on generalized Thullen domains

A comparison theorem of the Kobayashi metric and the Bergman metric is obtained on generalized Thullen domains in C 2 {{\mathbf {C}}^2} . This theorem is then used to obtain a lower estimate for the Kobayashi metric. It is noted that the lower estimate obtained is best possible.

Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings

A classical theorem of Carath6odory [8] states that every biholomorphic map f: D~ ~D2 between domains in the complex plane C bounded by simple closed Jordan curves extends to a homeomorphism of/31 onto/ )2 . There are some well-known generalizations of this result to domains in IE a. If D t and D2 are bounded pseudoconvex domains in 112" with if2 boundary and if the infinitezimal Kobayashi metric on D2 grows sufficiently fast near the boundary of D2(Ko2(g; X) >= [Xl/d(z, bDz) ~ for some e e (0, 1)), then every proper holomorphic map f : Dt--*D2 extends to a H61der-continuous map of/51 onto/32 [2, 3, 10, 25, 29-32]. This holds in particular if/92 is strictly pseudoconvex or if it is pseudoconvex with real-analytic boundary. The same result holds if D2 is only piecewise smooth strongly pseudoconvex [31]. Further results treat exceptional cases such as the balls [1 ], Reinhardt domains [5, 24], domains with many symmetries [4], and analytically bounded Hartogs domains in C 2 [I1]. Biholomorphic maps between certain types of nonpseudoconvex domains with real-analytic boundaries were treated in [13] and [27]. Besides the papers mentioned above there is a vast literature concerning smooth extension of proper holomorphic maps of smoothly bounded pseudoconvex domains ([6, 7, 12, 14], to mention just a few). In the present paper we obtain some results on the continuous extension of proper holomorphic maps f : D1--*/)2 under local assumptions on the boundaries bD1 and bD2. One of the main results is

Equivalence of the Classical Theorems of Schottky, Landau, Picard and Hyperbolicity

Modifying the classical theorems of Schottky and Landau, the author obtains the converses of these theorems. More precisely, the author defines the notions of Schottky, Landau and Picard properties and proves that a plane domain $D$ satisfies any of these properties if and only if ${\mathbf {C}}\backslash D$ contains at least two points. The method of proofs is completely elementary and uses only some basic properties of the Kobayashi metric.

Equivalence of the classical theorems of Schottky, Landau, Picard and hyperbolicity

Modifying the classical theorems of Schottky and Landau, the author obtains the converses of these theorems. More precisely, the author defines the notions of Schottky, Landau and Picard properties and proves that a plane domain D D satisfies any of these properties if and only if C ∖ D {\mathbf {C}}\backslash D contains at least two points. The method of proofs is completely elementary and uses only some basic properties of the Kobayashi metric.

Characterization of strictly convex domains biholomorphic to a circular domain

Recently parabolic exhaustions have been used successfully to classify complex spaces (Stoll [6 and 7], Burns [2], P. Wong [9]). Here we define parabolic exhaustions for strictly convex domains in C and give necessary and sufficient conditions for such a domain to be biholomorphically equivalent to a circular domain or even to the ball. The results presented in this note are part of the Ph.D. Thesis the author is completing under the direction of Professor W. Stoll. Details and more implications will appear at a later date. A strictly convex domain D C C is a domain for which there exists a defining function whose real Hessian is strictly positive on Tx(dD) for all x G dD. Let A be the open unit disk in C and let S be the unit sphere in C . Let D c C be a strictly convex domain and p E D be any point. For b G S Lempert [4] constructs an extremal map F(D, b) : A —• D which is holomorphic with F(0,6) = v and F'(0,6) = X where 1/X > 0 is the length of b in the infinitesimal Kobayashi metric of D at p. These conditions determine F(D, b) uniquely and the map extends smoothly to an embedding F(U, b) : S —• D. Also F: A x S —• D is of class C°° and surjective. One and only one function r : D -> R+ exists such that r(F(z,b)) = \z for (z,b) G A X S. Then r is a continuous exhaustion of D, positive and of class C°° on D {p}. Also r = 1 on 3D. We refer to r as the Lempert exhaustion of D at p.