Extrinsic curvature of codimension one isometric immersions with H\"older continuous derivatives

TL;DR

This paper demonstrates that for even-dimensional compact Riemannian manifolds with specific curvature properties, low-regularity isometric immersions have bounded extrinsic curvature, linking intrinsic and extrinsic curvature through an integral identity.

Contribution

It establishes a novel equivalence between intrinsic and extrinsic curvature for low-regularity isometric immersions in even dimensions, extending classical results to less smooth settings.

Findings

Extrinsic curvature equals intrinsic curvature for low-regularity immersions.

An integral identity for the Brouwer degree of the Gauss map is proven for low regularity.

Bounded extrinsic curvature surfaces are characterized under given conditions.

Abstract

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.