2016/01/22 by Sören Behr, Behr, Sören, Heiner Olbermann +1
Engineering · Mathematics · #26B10 #53A07 #53C24 #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #math.AP #math.DG #msc:26B10 #msc:53A07 #msc:53C24
paper · pdf · doi:10.48550/arxiv.1601.05959
20 pages; proof of Proposition 2 corrected
openalex publication_date 2016/01/22 · openalex created_date 2016/06/24 · arxiv created 2016/09/14 · arxiv updated 2016/09/15 · openalex updated_date 2026/07/28
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∈ C1,α(M;ℝn+1) is an isometric immersion with n/(n+1)< α≤ 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 C2 functions, but new for n>2 in the present context of low regularity.