Allard-type boundary regularity for $C^{1,\alpha}$ boundaries

TL;DR

This paper extends boundary monotonicity formulae for rectifiable varifolds to cases with $C^{1,eta}$ boundaries, enabling boundary regularity results under less smooth boundary conditions.

Contribution

It generalizes Allard's boundary monotonicity formulae from $C^{1,1}$ to $C^{1,eta}$ boundaries, broadening the scope of boundary regularity results for varifolds.

Findings

Boundary monotonicity formulae hold for $C^{1,eta}$ boundaries.

Boundary regularity results extend to less smooth boundaries.

Area ratios satisfy a monotonicity formula similar to interior cases.

Abstract

In this paper we show boundary monotonicity formulae for rectifiable varifolds having a $C^{1,\alpha}$ "boundary". In particular, we show that the area ratios of balls centered at this "boundary'' satisfy a nice monotonicity formula, similar to that for interior balls (proved in Allard's paper "On the first variation of a varifold''). This extends the boundary monotonicity formulae of Allard (see "On the first variation of a varifold- boundary behavior''), which require that the boundary is $C^{1,1}$. As a corollary, Allard's boundary regularity results extend to this case and provide a regularity result for rectifiable varifolds with a $C^{1,\alpha}$ ``boundary''.