We provide a sufficient condition for the continuous extension of isometries for the Kobayashi distance between bounded convex domains in complex Euclidean spaces having boundaries that are only slightly more regular than C 1 . This is a generalization of a recent result by A. Zimmer.
We investigate a form of visibility introduced recently by Bharali and Zimmerand shown to be possessed by a class of domains called Goldilocks domains. The range of theorems established for these domains stem from this form of visibility together with certain quantitative estimates that define Goldilocks domains. We show that some of the theorems alluded to follow merely from the latter notion of visibility. We call those domains that possess this property visibility domains with respect to the Kobayashi distance. We provide a sufficient condition for a domain in C n to be a visibility domain. A part of this paper is devoted to constructing a family of domains that are visibility domains with respect to the Kobayashi distance but are not Goldilocks domains. Our notion of visibility is reminiscent of uniform visibility in the context of CAT(0) spaces. However, this is an imperfect analogy because, given a bounded domain Ω in C n , n 2, it is, in general, not even known whether the metric space (Ω, kΩ) (where kΩ is the Kobayashi distance) is a geodesic space. Yet, with just this weak property, we establish two new Wolff-Denjoy-type theorems.2010 Mathematics Subject Classification. Primary: 32F45, 32H50, 53C23; Secondary: 32U05.
We make a connection between the structure of the bidisc and a distinguished subgroup of its automorphism group. The automorphism group of the bidisc, as we know, is of dimension six and acts transitively. We observe that it contains a subgroup that is isomorphic to the automorphism group of the open unit disc and this subgroup partitions the bidisc into a complex curve and a family of strongly pseudo-convex hypersurfaces that are non-spherical as CR-manifolds. Our work reverses this process and shows that any 2-dimensional Kobayashi-hyperbolic manifold whose automorphism group (which is known, from the general theory, to be a Lie group) has a 3-dimensional subgroup that is non-solvable (as a Lie group) and that acts on the manifold to produce a collection of orbits possessing essentially the characteristics of the concretely known collection of orbits mentioned above, is biholomorphic to the bidisc. The distinguished subgroup is interesting in its own right. It turns out that if we consider any subdomain of the bidisc that is a union of a proper sub-collection of the collection of orbits mentioned above, then the automorphism group of this subdomain can be expressed very simply in terms of this distinguished subgroup.
We study the action of the automorphism group of the 2 complex dimensional manifold symmetrized bidisc G on itself. The automorphism group is 3 real dimensional. It foliates G into leaves all of which are 3 real dimensional hypersurfaces except one, viz., the royal variety. This leads us to investigate Isaev's classification of all Kobayashi-hyperbolic 2 complex dimensional manifolds for which the group of holomorphic automorphisms has real dimension 3 studied by Isaev. Indeed, we produce a biholomorphism between the symmetrized bidisc and the domainIsaev calls it D1. The road to the biholomorphism is paved with various geometric insights about G.Several consequences of the biholomorphism follow including two new characterizations of the symmetrized bidisc and several new characterizations of D1. Among the results on D1, of particular interest is the fact that D1 is a "symmetrization". When we symmetrize (appropriately defined in the context in the last section) either Ω1 or D(2) 1 (Isaev's notation), we get D1. These two domains Ω1 and D(2) 1 are in Isaev's list and he mentioned that these are biholomorphic to D × D. We produce explicit biholomorphisms between these domains and D × D.
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