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Area minimising hypersurfaces mod $p$ do not admit immersed branch points

Paul Minter, Sidney Stanbury

Abstract

We show that area minimising hypersurfaces mod $p$ do not admit immersed branch points, namely branch points about which all classical singularities are immersed. Furthermore, we show that if an $n$-dimensional area minimising hypersurface mod $p$ is smoothly immersed outside a $\mathcal{H}^{n-1}$-null set, then it is in fact smoothly immersed outside a closed set of Hausdorff dimension at most $n-3$. These results are consequences of a more general analysis of immersed stable minimal hypersurfaces with a certain `alternating' orientation. Indeed, our proof does not rely on the minimising property other than through stationarity, stability, and the verification of simple structural properties of the hypersurface.

Area minimising hypersurfaces mod $p$ do not admit immersed branch points

Abstract

We show that area minimising hypersurfaces mod do not admit immersed branch points, namely branch points about which all classical singularities are immersed. Furthermore, we show that if an -dimensional area minimising hypersurface mod is smoothly immersed outside a -null set, then it is in fact smoothly immersed outside a closed set of Hausdorff dimension at most . These results are consequences of a more general analysis of immersed stable minimal hypersurfaces with a certain `alternating' orientation. Indeed, our proof does not rely on the minimising property other than through stationarity, stability, and the verification of simple structural properties of the hypersurface.
Paper Structure (10 sections, 15 theorems, 45 equations, 2 figures)

This paper contains 10 sections, 15 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction & Main Results
  2. Background
  3. Main Results for $\mathcal{S}_Q^\pm$
  4. Proof of Theorem \ref{['thm:eps-reg']} ($\varepsilon$-Regularity)
  5. $C^{1,\alpha}$ Selections
  6. $C^{1,1}$ Regularity via Planar Frequency
  7. Proof of Theorem \ref{['thm:eps-reg']}
  8. Proof of Partial Regularity & Compactness
  9. The Non-Immersed Case & Open Questions
  10. Results for $\mathcal{S}_Q$

Key Result

Theorem 1.1

Let $T^n$ be an area minimising hypersurface mod $p$ in $B^{n+1}_2(0)$ with $\partial^p T = 0$. Suppose all classical singularities of $T$ are immersed. Then, $T$ does not have any branch points.

Figures (2)

  • Figure 1: If one takes $p\geq 3$ distinct points (positively oriented) on a circle in $\mathbb{R}^2$, an area minimiser mod $p$ with this boundary will always contain a classical singularity (of density $p/2$) in the interior. If $p\geq 6$ is even and the points are in general position, there will always be a non-immersed classical singularity (as generically $p/2$ ($\geq 3$) lines do not intersect at a common point).
  • Figure 2: An area minimiser mod $4$ with a flat singular point (the tangent plane has multiplicity $2$). Note that this is not a branch point, as the surface decomposes as Enneper's surface with a tangent plane. The fact that this surface is area minimising mod $4$ follows from White79.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5: The Riemannian Case
  • Remark 1.6
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 22 more