The automorphism group and limit set of a bounded domain I: the finite type case

Abstract

For bounded pseudoconvex domains with finite type we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different points of the boundary, then the automorphism group has finitely many components and is the almost direct product of a compact group and connected Lie group locally isomorphic to $\mathrm{Aut}({\mathbb{B}}_k)$. Further, the limit set is a smooth submanifold diffeomorphic to the sphere of dimension $2k-1$. As applications we prove a new finite jet determination theorem and a Tits alternative theorem. The geometry of the Kobayashi metric plays an important role in the paper.

arXiv comment33 pages. v3: minor revisions. Final version to appear in Advances in Mathematics