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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Intrinsic curvature of the induced metric on harmonically immersed surfaces


Author: Tilla Klotz Milnor
Journal: Proc. Amer. Math. Soc. 94 (1985), 549-552
MSC: Primary 53C50; Secondary 53C42, 58E20
MathSciNet review: 787911
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Abstract: A theorem by Wissler is used to prove the following result. Suppose that an oriented surface $ S$ with indefinite prescribed metric $ h$ is harmonically mapped into an arbitrary pseudo-Riemannian manifold so that the metric $ I$ induced on $ S$ is complete and Riemannian. Then the intrinsic curvature $ K\left( I \right)$ of the immersion satisfies $ {\text{inf}}\left\vert {K\left( I \right)} \right\vert = 0$, with $ {\text{sup}}\left\vert {{\text{grad 1/K}}\left( I \right) = \infty } \right.$ in case $ K\left( I \right)$ never vanishes on $ S$.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0787911-4
Article copyright: © Copyright 1985 American Mathematical Society