Decay estimates for the quadratic tilt-excess of integral varifolds

TL;DR

This paper establishes optimal pointwise decay estimates for the quadratic tilt-excess of integral varifolds with first variation in Lebesgue spaces or Radon measures, advancing understanding of their geometric regularity.

Contribution

It provides new optimal decay estimates for the quadratic tilt-excess of integral varifolds under broad conditions, including arbitrary dimensions and Lebesgue space exponents.

Findings

Optimal decay rates for quadratic tilt-excess established

Results apply to varifolds with first variation in Lebesgue spaces or Radon measures

Enhanced understanding of geometric regularity of varifolds

Abstract

This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space with its first variation given by either a Radon measure or a function in some Lebesgue space. Pointwise decay results for the quadratic tilt-excess are established for those varifolds. The results are optimal in terms of the dimension of the varifold and the exponent of the Lebesgue space in most cases, for example if the varifold is not two-dimensional.