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Intrinsic Curvature of the Induced Metric on Harmonically Immersed Surfaces

Tilla Klotz Milnor
Proceedings of the American Mathematical Society
Vol. 94, No. 3 (Jul., 1985), pp. 549-552
DOI: 10.2307/2045252
Stable URL: http://www.jstor.org/stable/2045252
Page Count: 4
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Intrinsic Curvature of the Induced Metric on Harmonically Immersed Surfaces
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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(I) of the immersion satisfies $\inf|K(I)| = 0$, with $\sup|\operatorname{grad} 1/K(I) = \infty$ in case K(I) never vanishes on S.

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