October  1999, 5(4): 799-804. doi: 10.3934/dcds.1999.5.799

Harmonic maps on complete manifolds

1. 

Department of Mathematics, Southwest Missouri State University

2. 

Department of Applied Mathematics, University of Colorado at Boulder

Received  November 1998 Revised  November 1998 Published  July 1999

In this article, we study harmonic maps between two complete noncompact manifolds M and N by a heat flow method. We find some new sufficient conditions for the uniform convergence of the heat flow, and hence the existence of harmonic maps.
Our condition are: The Ricci curvature of M is bounded from below by a negative constant, M admits a positive Green’s function and

$ \int_M G(x, y)|\tau(h(y))|dV_y $ is bounded on each compact subset. $\qquad$ (1)

Here $\tau(h(x))$ is the tension field of the initial data $h(x)$.
Condition (1) is somewhat sharp as is shown by examples in the paper.

Citation: Wenxiong Chen, Congming Li. Harmonic maps on complete manifolds. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 799-804. doi: 10.3934/dcds.1999.5.799
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