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Harmonic maps on complete manifolds

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  • 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.

    Mathematics Subject Classification: 35J60, 53C99.


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