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Minimal Surfaces with Stratified Branching Sets

Federico Franceschini, Rafe Mazzeo, Paul Minter

Abstract

Inspired by the Taubes-Wu construction of $\mathcal{C}^{1,α}$ two-valued harmonic functions by the use of symmetry, we construct minimal surfaces with stratified branching sets as graphs of $\mathcal{C}^{1,α}$ 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.

Minimal Surfaces with Stratified Branching Sets

Abstract

Inspired by the Taubes-Wu construction of two-valued harmonic functions by the use of symmetry, we construct minimal surfaces with stratified branching sets as graphs of 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 -ball or, slightly more generally, over the product of with a torus , 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 -ball or , 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 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.

Paper Structure

This paper contains 15 sections, 21 theorems, 80 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Comments
  5. Acknowledgements
  6. Perturbative Examples
  7. Elliptic conic theory and the Caffarelli--Hardt--Simon theorem
  8. Eigenfunctions on $S^2$ with branch points
  9. An extension to cylinders
  10. Minimal submanifolds with stratified branching set
  11. Branched minimal hypersurfaces with large boundary data
  12. Setup
  13. Existence of minimal surfaces in convex polyhedra
  14. Proof of Theorem \ref{['thm:main']}
  15. Compact Examples

Key Result

Proposition 2.3

Given the conic manifold $X$, let $\Gamma$ denote the union of the sets of indicial roots $\{\gamma_j^\pm\}$ over all conic points of $X$. For simplicity, assume all $\gamma_j^+$ lie in $(-k/2, +\infty)$. Then for any $\mu \in \mathbb R \setminus \Gamma$, the map is Fredholm. When $X$ is the exact cone $C$, this map is injective when $\mu>\gamma_1^-$ and surjective when $\mu<\gamma_1^+$, and henc

Figures (3)

  • Figure 1: Plot of $\text{Re}(x+iy)^{3/2}.$ The graph cannot be disentangled into two distinct smooth graphs around the origin: it is a branch point.
  • Figure 2: The spherical tetrahedron, which is determined by the spherical projection of the standard 3-simplex in $\mathbb{R}^3$.
  • Figure 3: The bifurcation argument for $S^1\times(-1,1)$ and $\Gamma=\mathbb{Z}_3$. On the left $\lambda$ is small and there is only one geodesic connecting the pink points, namely an equator. On the right $\lambda$ is large, so the equator becomes unstable and two stable geodesics appear (in dark green).

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: Maz
  • Proposition 2.4: Maz
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Proposition 2.7
  • proof
  • Theorem 2.8
  • ...and 29 more