Characterizing domains by the limit set of their automorphism group

Abstract

Abstract. In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a "polynomial ellipsoid" (that is, a domain defined by a weighted homogeneous balanced polynomial) if and only if the limit set of the automorphism group intersects at least two closed complex faces of the set. The proof relies on a detailed study of the geometry of the Kobayashi metric and ideas from the theory of non-positively curved metric spaces. We also obtain a number of other results including the Greene-Krantz conjecture in the case of uniform non-tangential convergence, new results about continuous extensions (of biholomorphisms and complex geodesics), and a new Wolff-Denjoy theorem.

On the continuous extension of Kobayashi isometries

“…Thus, we may suppose that ǫ 0 is so small that, for every y ∈ (−ǫ 0 , ǫ 0 ), αω(|y|) 1. Therefore, for an arbitrary s ∈ (−G(ǫ 0 ), G(ǫ 0 )), We are now ready to state and prove a generalization of Proposition 4.3 in [13]. The generalization of the latter result alone suffices to yield a generalization of Theorem 2.11 in [13], which is fundamental to establishing an extension-of-isometries theorem.…”

“…The goal of this section is to prove certain technical results that are essential for extending the scope of an idea in [13] to the sorts of domains considered in Theorem 1.3. Specifically: that inward-pointing normals can be parametrized as K-almost-geodesics for some K > 0.…”

“…The earliest result in this direction was given by Lempert [8], which states that if Ω ⊂ C n is strongly convex with C k -smooth boundary, k 2, then every complex geodesic F : D → Ω extends to a C k−2 -smooth mapping on D (by a C 0 -smooth mapping we mean a continuous one). Since then, there has been a number of works dealing with the continuous (or smooth) extension of complex geodesics; see [1,10,2,13].…”

“…Before we state the main result of this paper, let us look at the motivations behind it. Our chief motivation is the following recent result by Zimmer: [13,Theorem 2.18]). Let Ω j ⊂ C n j , j = 1, 2, be bounded convex domains with C 1,α -smooth boundaries, where α ∈ (0, 1).…”

A weak notion of visibility, a family of examples, and Wolff-Denjoy theorems

“…For various reasons having to do with their intrinsic geometry, convex domains predominate among recent generalizations of the Wolff-Denjoy theorem: see, for instance, [8,11,4,29] and several of the results in [20]. Visibility in the sense of Definition 1.1 is one of the key ingredients in the proof by Bharali-Zimmer of a generalization [9, Theorem 1.10] of Result 1.7 to taut Goldilocks domains.…”