On the rigidity and boundary regularity for Bakry-Emery-Kohn harmonic functions in Bergman metric on the unit ball in $\textit {C}^n$

Abstract:

In this paper we study the rigidity theorem for smooth Bakry-Emery-Kohn harmonic function $u$ in the unit ball $B_n$ in $\textit {C}^n$, which satisfies \begin{equation*} \Box ^{\psi } u=\sum _{i, j=1}^n (\delta _{ij}-z_i \overline {z}_j){\partial ^2 u \over \partial z_i \partial \overline {z}_j}+(1-|z|^2) \psi ’(|z|^2) \sum _{j=1}^n \overline {z}_j {\partial u\over \partial \overline {z}_j}=0 \end{equation*} with some restriction of the coefficients of Taylor expansion for $\psi$ at $1$. We prove that any smooth B-E-K harmonic function on $\overline {B}_n$ must be holomorphic in $B_n$. We study the regularity problem for the solution of the Dirichlet boundary value problem: \begin{equation*} \begin {cases} \Box ^\psi u=0, \hbox { if } z\in B_n,\cr \quad \ u=f, \hbox { if } z\in \partial B_n.\cr \end{cases} \end{equation*}