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Compactness of certain class of singular minimal hypersurfaces

Akashdeep Dey

Abstract

Given a closed Riemannian manifold $(N^{n+1},g)$, $n+1 \geq 3$ we prove the compactness of the space of singular, minimal hypersurfaces in $N$ whose volumes are uniformly bounded from above and the $p$-th Jacobi eigenvalue $λ_p$'s are uniformly bounded from below. This generalizes the results of Sharp and Ambrozio-Carlotto-Sharp in higher dimensions.

Compactness of certain class of singular minimal hypersurfaces

Abstract

Given a closed Riemannian manifold , we prove the compactness of the space of singular, minimal hypersurfaces in whose volumes are uniformly bounded from above and the -th Jacobi eigenvalue 's are uniformly bounded from below. This generalizes the results of Sharp and Ambrozio-Carlotto-Sharp in higher dimensions.

Paper Structure

This paper contains 8 sections, 12 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Notations
  4. Preliminaries from geometric measure theory
  5. Index and Jacobi eigenvalues of a stationary varifold
  6. Modifications of the results of Schoen-Simon SS
  7. Proof of the main theorem
  8. Some further remarks

Key Result

Theorem \oldthetheorem

Let $(N^{n+1},g)$ be an arbitrary closed Riemannian manifold with $n+1 \geq 3$. Then $N$ contains a singular, minimal hypersurface which is smooth and embedded outside a singular set of Hausdorff dimension atmost $n-7$. In particular, when $3 \leq n+1 \leq 7$ there exists a smooth, closed, embedded

Theorems & Definitions (27)

  • Theorem \oldthetheorem: Alm, Pitts, SS
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: MN_index
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • ...and 17 more