Asymptotic Dirichlet problem for harmonic maps on negatively curved manifolds

Seok Woo Kim; Yong Hah Lee

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2005; 42(3): 543-553

Printed May 1, 2005

Asymptotic Dirichlet problem for harmonic maps on negatively curved manifolds

Seok Woo Kim and Yong Hah Lee

Konkuk University, Ewha Womans University

Abstract

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Abstract

In this paper, we prove the existence of nonconstant bounded harmonic maps on a Cartan-Hadamard manifold of pinched negative curvature by solving the asymptotic Dirichlet problem. To be precise, given any continuous data $f$ on the boundary at infinity with image within a ball in the normal range, we prove that there exists a unique harmonic map from the manifold into the ball with boundary value $f$.

Keywords: asymptotic Dirichlet Problem, harmonic maps

MSC numbers: 58E20, 31B05