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Rigidity theorem for harmonic maps with complex normal boundary conditions

Song-Ying Li

Department of Mathematics, University of California, Irvine, Irvine, CA 92697–3875, USA

E-mail Address: sli@math.uci.edu

Corresponding author.

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Jie Luo

College of Mathematics and Informatics, Fujian Normal University, Fuzhou, Fujian 350117, PRC

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PRC

E-mail Address: jluo@amss.ac.cn

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https://doi.org/10.1142/S0129167X19400019Cited by:0 (Source: Crossref)

This article is part of the issue:

Special Issue on Geometric Analysis and General Relativity

Abstract

Let $B_{n}$ $B_{n}$ be the open unit ball in $ℂ^{n}$ and let $(M^{m}, g)$ be a Kähler manifold with strongly negative or strongly semi-negative curvature. In this paper, we study Siu type rigidity theorem for the harmonic map $u \in C^{2} (\bar{B_{n}}, M)$ satisfying the boundary condition that $\sum_{i, j = 1}^{n} z_{i} {\bar{z}}_{j} u_{i \bar{j}} = 0$ on $\partial B_{n}$ . We also prove the existence and uniqueness theorem for some Neumann type boundary value problem for harmonic functions on $B_{n}$ .

Keywords:

AMSC: 32W25, 32W50

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