A rigidity result for domains with a locally strictly convex point

Abstract

In this article, we investigate projective domains with a strictly convex point in the boundary and their automorphisms. We prove that ellipsoids can be characterized as follows: A domain Ω is an ellipsoid if and only if ∂Ω is locally strongly convex at some boundary point where an Aut(Ω)-orbit accumulates. We also show that every quasi-homogeneous projective domain in an affine space which is locally strictly convex at a boundary point, is the universal covering of a closed projective manifold.

Characterizing the unit ball by its projective automorphism group

“…Colbois and Verovic [CV02] gave an alternative proof with the additional assumption that ∂Ω is C 3 . Later Jo [Jo08] and Yi [Yi08] proved that it is enough to assume that L(Ω) contains a point x where ∂Ω is strongly convex in a neighborhood of x.…”