Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds

TL;DR

This paper establishes precise decay rates for the quadratic tilt-excess of integral varifolds, provides counter-examples for super-quadratic tilt-excess decay, and explores implications for curvature varifolds and integrability conditions.

Contribution

It precisely characterizes decay rates of tilt-excess for 2D integral varifolds and constructs optimal counter-examples for super-quadratic cases, advancing understanding of varifold regularity.

Findings

Exact decay rate for quadratic tilt-excess in 2D varifolds.

Counter-examples showing limits of decay rates for super-quadratic tilt-excess.

Demonstration that bounded mean curvature does not imply higher integrability of second fundamental form.

Abstract

This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. In order to obtain the exact decay rate, a coercive estimate involving a height-excess quantity measured in Orlicz spaces is established. Moreover, counter-examples to pointwise power decay rates almost everywhere of the super-quadratic tilt-excess are obtained. These examples are optimal in terms of the dimension of the varifold and the exponent of the integrability condition in most cases, for example if the varifold is not two-dimensional. These examples also demonstrate that within the scale of Lebesgue spaces no local higher integrability of the second fundamental form, of an at least two-dimensional curvature varifold, may be deduced from boundedness of its generalised mean curvature vector. Amongst the tools are Cartesian products of curvature varifolds.