# American Institute of Mathematical Sciences

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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