Intrinsic curvature of the induced metric on harmonically immersed surfaces

Milnor, Tilla Klotz

doi:10.1090/S0002-9939-1985-0787911-4

Intrinsic curvature of the induced metric on harmonically immersed surfaces
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Proc. Amer. Math. Soc. 94 (1985), 549-552 Request permission

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 ${\text {inf}}\left | {K\left ( I \right )} \right | = 0$, with ${\text {sup}}\left | {{\text {grad 1/K}}\left ( I \right ) = \infty } \right .$ in case $K\left ( I \right )$ never vanishes on $S$.

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