Minimal Surfaces with Stratified Branching Sets

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.