Dynamics of Local Automorphisms of Embedded CR-Manifolds

Abstract

A classical theorem due to Wong [1] states that the ball is the unique strictly pseudoconvex bounded domain having a noncompact automorphism group. Rosay [2] extended this theorem to bounded domains such that an orbit accumulates at a strictly pseudoconvex domain at the boundary. Later, Efimov [3] got rid of the boundedness assumption. Schoen [4] and, independently, Spiro [5] proved the CR-version of this result, where the strict pseudoconvexity of the accumulation point of an orbit also plays a crucial role. In the present note, we suggest two local CR-versions of this result for hypersurfaces. These versions manifest different behavior depending on whether the Levi form is definite or not. Let M be a real-analytic Levi-nondegenerate hypersurface in some neighborhood U of the origin in C , n ≥ 1 . Denote by Aut0 M the group of local diffeomorphisms of M , that is, the group of germs of biholomorphisms taking M locally into itself and preserving the origin. A point q = 0 (treated as its own radius-vector −→ 0q) is said to be contractible if there is a sequence of local automorphisms Φn ∈ Aut0 M such that