A novel type of Sobolev-Poincare inequality for submanifolds of Euclidean space

Menne, Ulrich; Scharrer, Christian

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A novel type of Sobolev-Poincare inequality for submanifolds of Euclidean space

MPS-Authors

Scharrer, Christian
AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Menne, U., & Scharrer, C. (submitted). A novel type of Sobolev-Poincare inequality for submanifolds of Euclidean space.

Cite as: https://hdl.handle.net/11858/00-001M-0000-002E-07B5-8

Abstract

For functions on generalised connected surfaces (of any dimensions) with
boundary and mean curvature, we establish an oscillation estimate in which the
mean curvature enters in a novel way. As application we prove an a priori
estimate of the geodesic diameter of compact connected smooth immersions in
terms of their boundary data and mean curvature. These results are developed in
the framework of varifolds. For this purpose, we establish that the notion of
indecomposability is the appropriate substitute for connectedness and that it
has a strong regularising effect; we thus obtain a new natural class of
varifolds to study. Finally, our development leads to a variety of questions
that are of substance both in the smooth and the nonsmooth setting.