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Singularity removal rigidity theorems for minimal hypersurfaces in manifolds with nonnegative scalar curvature

Shihang He, Yuguang Shi, Haobin Yu

Abstract

We prove two "Singularity removal rigidity theorems" for minimal hypersurfaces with isolated singularities in manifolds of nonnegative scalar curvature (Theorems \ref{thm: rigidity for minimal surface} and \ref{thm: georch free of singularity}). In particular, we observe a new phenomenon that the extremal scalar curvature condition forces smoothness, which reveals a kind of positive effect of minimal hypersurface singularities in scalar curvature geometry. As an application, we obtain a direct proof of the positive mass theorem (PMT) for asymptotically flat $8$-manifolds with arbitrary ends (Theorem \ref{thm: pmt8dim}), without using N. Smale's generic regularity theorem. A key ingredient is a new spectral version of PMT for AF manifolds with arbitrary ends, whose proof relies on PMT for asymptotically locally flat (ALF) manifolds with $\mathbf{S}^1$-symmetry.

Singularity removal rigidity theorems for minimal hypersurfaces in manifolds with nonnegative scalar curvature

Abstract

We prove two "Singularity removal rigidity theorems" for minimal hypersurfaces with isolated singularities in manifolds of nonnegative scalar curvature (Theorems \ref{thm: rigidity for minimal surface} and \ref{thm: georch free of singularity}). In particular, we observe a new phenomenon that the extremal scalar curvature condition forces smoothness, which reveals a kind of positive effect of minimal hypersurface singularities in scalar curvature geometry. As an application, we obtain a direct proof of the positive mass theorem (PMT) for asymptotically flat -manifolds with arbitrary ends (Theorem \ref{thm: pmt8dim}), without using N. Smale's generic regularity theorem. A key ingredient is a new spectral version of PMT for AF manifolds with arbitrary ends, whose proof relies on PMT for asymptotically locally flat (ALF) manifolds with -symmetry.
Paper Structure (21 sections, 45 theorems, 203 equations, 5 figures)

This paper contains 21 sections, 45 theorems, 203 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Singularity removal rigidity theorems and Geroch's Conjecture in Dimensions Not Exceeding $8$
  3. Positive mass theorem on AF manifolds with arbitrary ends and nonnegative scalar curvature in the spectral sense.
  4. Outline of Proofs
  5. Organization of The Paper
  6. Preliminaries
  7. Asymptotically locally flat manifolds
  8. PSC manifolds with free $\mathbf{S}^1$ actions
  9. Singular harmonic functions on area-minimizing hypersurfaces with isolated singularities
  10. Sobolev space on metric measure spaces
  11. Singular harmonic functions on a compact domain
  12. Extension of the singular harmonic function to the total space
  13. Positive mass theorem for AF manifolds with arbitrary ends and PSC in the strong spectral sense
  14. Deformation and warped construction of minimal hypersurfaces with singular set
  15. Blow up-warped construction of area-minimizing hypersurface with singularities in asymptotically flat manifold
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $(M^{n+1},g)$ be an AF manifold with arbitrary ends, an AF end $E$ of asymptotic order $\tau>n-2$. Let $\Sigma^{n}\subset M^{n+1}$ be an area-minimizing boundary in $M^{n+1}$ with isolated singular set $\mathcal{S}$. Assume $\Sigma$ is strongly stable (see Definition defn: strongly stable hypers

Figures (5)

  • Figure 1: A schematic illustration of the proof of Proposition \ref{['prop: eq1-arbitrary end']}
  • Figure 2: The "blow up - warped product" procedure
  • Figure 3: free boundary problems with inner obstacle
  • Figure 4: A noncompact manifold that admits a non-zero degree map to an enlargeable manifold
  • Figure 5: The illustration of the band $\hat{\mathcal{V}}$ obtained from the pullback construction

Theorems & Definitions (103)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 93 more