A Sobolev Poincar\'e type inequality for integral varifolds

TL;DR

This paper establishes a local Sobolev Poincaré type inequality for integral varifolds, linking their deviation from a multivalued plane to tilt and curvature, with sharp bounds on Lebesgue exponents.

Contribution

It introduces a novel inequality that bounds the distance of integral varifolds from a multivalued plane using tilt and mean curvature, with optimal exponent bounds.

Findings

Inequality bounds the varifold's deviation by tilt and curvature

Exponents in Lebesgue spaces are proven to be sharp

Provides a new analytical tool for studying varifolds

Abstract

In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.