Harmonic maps on complete manifolds

Wenxiong Chen; Congming Li

doi:10.3934/dcds.1999.5.799

Article Contents

1999, Volume 5, Issue 4: 799-804. Doi: 10.3934/dcds.1999.5.799

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Harmonic maps on complete manifolds

Wenxiong Chen^1, and
Congming Li^2,

1.
Department of Mathematics, Southwest Missouri State University
2.
Department of Applied Mathematics, University of Colorado at Boulder

Received: October 31, 1998

Revised: October 31, 1998

Published: September 1999

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Abstract

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.

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Mathematics Subject Classification: 35J60, 53C99.

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