The regularity of parametrized integer stationary varifolds in two dimensions

TL;DR

This paper proves that parametrized two-dimensional stationary varifolds are smooth minimal branched immersions with constant multiplicity, advancing understanding in geometric measure theory and variational calculus.

Contribution

It establishes an optimal regularity result showing parametrized stationary varifolds are smooth minimal branched immersions with constant multiplicity in two dimensions.

Findings

Parametrized stationary varifolds are smooth minimal branched immersions.

The multiplicity function is shown to be constant.

Applications in the calculus of variations for the area functional.

Abstract

We establish an optimal regularity result for parametrized two-dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant. We provide some applications of this regularity result, especially in the calculus of variations for the area functional.