On the rigidity and boundary regularity for Bakry-Emery-Kohn harmonic functions in Bergman metric on the unit ball in 𝐶ⁿ

Abstract

In this paper we study the rigidity theorem for smooth Bakry-Emery-Kohn harmonic function u in the unit ball B n in C n , which satisfieswith some restriction of the coefficients of Taylor expansion for ψ at 1. We prove that any smooth B-E-K harmonic function on B n must be holomorphic in B n . We study the regularity problem for the solution of the Dirichlet boundary value problem: