A frequency function and singular set bounds for branched minimal immersions

TL;DR

This paper proves regularity and bounds on the singular set of 2-valued stationary functions related to branched minimal immersions, establishing optimal regularity and dimensional estimates for branch points.

Contribution

It introduces bounds on the frequency function and singular set for 2-valued C^{1, α} functions, advancing understanding of branched minimal immersions.

Findings

2-valued C^{1, 1/2} regularity is established.

The singular set has Hausdorff dimension at most (n-2).

The regularity result is shown to be optimal.

Abstract

We show that any 2-valued C^{1, \alpha} (\alpha \in (0, 1)) function u = {u_{1}, u_{2}} on an open ball B in {\mathbb R}^{n} with values u_{1}, u_{2} \in {\mathbb R}^{k} whose graph, viewed as a varifold with multiplicity 2 at points where u_{1} = u_{2} and with multiplicity 1 at points where u_{1}, u_{2} are distinct, is stationary in the cylinder B \times {\mathbb R}^{k} must be a C^{1, 1/2} function, and the set of its branch points, if non-empty, must have Hausdorff dimension (n-2) and locally positive (n-2)-dimensional Hausdorff measure. The C^{1, 1/2} regularity is optimal.