The Regularity of Conformal Target Harmonic Maps

TL;DR

This paper proves that weakly conformal $W^{1,2}$ maps from a Riemann surface into a submanifold are smooth conformal harmonic maps if and only if they are stationary varifolds, establishing a regularity result in geometric analysis.

Contribution

It establishes a regularity criterion linking stationary varifolds to smooth conformal harmonic maps for weakly conformal $W^{1,2}$ maps.

Findings

Weakly conformal $W^{1,2}$ maps are smooth conformal harmonic maps if stationary.

Characterization of regularity for conformal harmonic maps.

Connection between stationary varifolds and smoothness of maps.

Abstract

In a recent paper the author introduced a new method based on viscosity techniques for producing minimal surfaces by minmax arguments. The present work corresponds to the regularity part of the method. Precisely we establish that any weakly conformal $W^{1,2}$ map from a riemann surface $S$ into a closed oriented sub-manifold $N^n$ of an euclidian space ${\mathbb R}^m$ realizes a stationary varifold if and only if it is a smooth conformal harmonic map form $S$ into $N^n$.