Minimal Surfaces with Stratified Branching Sets

TL;DR

This paper constructs minimal surfaces with stratified branching sets using three methods, expanding the class of known minimal submanifolds with complex branching structures and novel frequency properties.

Contribution

It introduces three new constructions of minimal surfaces with stratified branching sets, utilizing perturbation, barrier, and bifurcation techniques.

Findings

Constructed branched minimal submanifolds in arbitrary codimension.

Produced stable minimal hypersurfaces with large boundary data.

Generated compact minimal submanifolds with non-trivial stratified branching sets.

Abstract

Inspired by the Taubes-Wu construction of $\mathcal{C}^{1,\alpha}$ two-valued harmonic functions by the use of symmetry, we construct minimal surfaces with stratified branching sets as graphs of $\mathcal{C}^{1,\alpha}$ two-valued functions. We give three constructions. The first is perturbative and produces branched minimal submanifolds in arbitrary codimension as two-valued graphs over the $n$-ball or, slightly more generally, over the product of $B^n$ with a torus $\mathbb T^N$, parametrized by boundary data which is required to be small in a suitable norm. The second uses barrier methods together with a reflection argument to produce branched stable minimal hypersurfaces, again as two-valued graphs over the unit $n$-ball or $B^n \times \mathbb T^N$, parametrized by boundary data which now can be large. Finally, using bifurcation theory, we produce compact minimal submanifolds with similarly stratified branching sets in an ambient space $S^n \times \mathbb R$ with a suitable (analytic) warped product metric. These examples give minimal submanifolds with novel frequency values and whose branching sets have non-trivial deeper strata. While the main constructions are fairly elementary, they rely on the use of precisely tailored (and somewhat non-standard) function spaces, combined with a regularity theory which provides full asymptotic expansions around the branching sets.