On univalent harmonic maps between surfaces

Richard Schoen Shing-Tung Yau

Differential Geometry mathscidoc:1912.43478

Inventiones mathematicae, 44, (3), 265-278, 1978.10
Hence the energy defines a functional on the space of Lipshitz maps between M and M'. Critical points of this functional are called harmonic maps. These maps were studied by Bochner, Morrey, Rauch, Eells and Sampson, Hartman, Uhlenbeck, Hamilton, Hildebrandt and others. The first fundamental result was due to Eells and Sampson [3] who proved that, in case M'has non-positive sectional curvature, each map from M to M'is homotopic to a harmonic map.(This result was then extended by Hamilton [8] to the case where both M and M'are allowed to have boundary.) Later Hartman [7] was able to prove the harmonic map is unique in each homotopy class if M'has strictly negative curvature. This last result of Hartman leads one to believe that harmonic maps between compact manifolds with negative curvature must enjoy a lot of nice properties. In fact, a few years ago, B. Lawson and the second author conjectured
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  title={On univalent harmonic maps between surfaces},
  author={Richard Schoen, and Shing-Tung Yau},
  booktitle={Inventiones mathematicae},
Richard Schoen, and Shing-Tung Yau. On univalent harmonic maps between surfaces. 1978. Vol. 44. In Inventiones mathematicae. pp.265-278. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203516243640042.
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