Calculus of Variations and Geometric Measure Theory
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S. Behr - H. Olbermann

Extrinsic curvature of codimension one isometric immersions with Hölder continuous derivatives

created by olbermann on 23 Jul 2019



Inserted: 23 jul 2019

Year: 2016

ArXiv: 1601.05959 PDF


We prove that if $n$ is even, $(M,g)$ is a compact $n$-dimensional Riemannian manifold whose Pfaffian form is a positive multiple of the volume form, and $y\in C^{1,\alpha}(M;\mathbb{R}^{n+1})$ is an isometric immersion with $n/(n+1)< \alpha\leq 1$, then $y(M)$ is a surface of bounded extrinsic curvature. This is proved by showing that extrinsic curvature, defined by a suitable pull-back of the volume form on the $n$-sphere via the Gauss map, is identical to intrinsic curvature, defined by the Pfaffian form. This latter fact is stated in form of an integral identity for the Brouwer degree of the Gauss map, that is classical for $C^2$ functions, but new for $n>2$ in the present context of low regularity.

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