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Generic Scarring for Minimal Hypersurfaces in Manifolds Thick at Infinity with a Thin Foliation at Infinity

Xingzhe Li

Abstract

We show generic scarring phenomenon for minimal hypersurfaces in a class of complete non-compact manifolds. In particular, we prove that for any metric $g$ in a $C^{\infty}$-generic subset of the family of complete metrics which are thick at infinity with a thin foliation at infinity on a fixed $M^{n+1}$ of dimension $3 \leq (n + 1) \leq 7$, to any connected, closed, embedded, $2$-sided, stable minimal hypersurface $S \subset (M, g)$, there exists a sequence of closed, embedded, minimal hypersurfaces $\{Σ_{k}\}$ scarring along $S$, in the sense that the area of $Σ_{k}$ diverges to infinity, and when properly renormalized, $Σ_{k}$ converges to $S$ as varifolds.

Generic Scarring for Minimal Hypersurfaces in Manifolds Thick at Infinity with a Thin Foliation at Infinity

Abstract

We show generic scarring phenomenon for minimal hypersurfaces in a class of complete non-compact manifolds. In particular, we prove that for any metric in a -generic subset of the family of complete metrics which are thick at infinity with a thin foliation at infinity on a fixed of dimension , to any connected, closed, embedded, -sided, stable minimal hypersurface , there exists a sequence of closed, embedded, minimal hypersurfaces scarring along , in the sense that the area of diverges to infinity, and when properly renormalized, converges to as varifolds.
Paper Structure (9 sections, 12 theorems, 79 equations, 1 figure)

This paper contains 9 sections, 12 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Single-Cylindrical Width and Associated Min-max Theory
  3. Min-max theory in compact manifolds with boundary
  4. Uniform bound of single-cylindrical widths
  5. Single-cylindrical widths are locally Lipschitz functions of the metric
  6. A quantitative constancy theorem
  7. Metric deformation and approximation by minimal hypersurfaces
  8. Generic Scarring for minimal hypersurfaces along stable hypersurfaces
  9. Appendix: Proof of Theorem \ref{['qctc']}

Key Result

Theorem 1.1

Let $M^{n + 1}$ be a smooth manifold of dimension $3 \leq (n + 1) \leq 7$. For any metric $g$ in a $C^{\infty}$-generic subset of $\mathcal{F}_{\mathop{\mathrm{thin}}\nolimits} \cap \mathop{\mathrm{Int}}\nolimits(\mathcal{T}_{\infty})$ in the sense of Baire, the following holds. For any connected, c

Figures (1)

  • Figure 1: A schematic illustration of a sequence of minimal hypersurfaces $\{\tilde{\Gamma}_{m, p_{k}}\}$.

Theorems & Definitions (26)

  • Theorem 1.1: Main Theorem
  • Remark
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Song19
  • Definition 2.1
  • Theorem 2.2: Single-Cylindrical Weyl Law
  • Definition 2.3
  • Lemma 2.4: Sun-Wang-Zhou20
  • Theorem 2.5
  • ...and 16 more