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Examples of compact embedded mean convex $λ$-hypersurfaces

Qing-Ming Cheng, Junqi Lai, Guoxin Wei

Abstract

There is a well-known conjecture asserts that the round sphere should be the only compact embedded self-shrinker (i.e. $0$-hypersurface) which is diffeomorphic to a sphere. S. Brendle confirmed the conjecture for 2-dimensional $0$-hypersurfaces. For any dimensional $λ$-hypersurfaces, if $λ<0$, we constructed compact convex embedded $λ$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In this paper, for $λ>0$, we construct a compact mean convex embedded $λ$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In fact, for $λ>0$, there are no compact convex embedded $λ$-hypersurfaces which are diffeomorphic to spheres except a round sphere.

Examples of compact embedded mean convex $λ$-hypersurfaces

Abstract

There is a well-known conjecture asserts that the round sphere should be the only compact embedded self-shrinker (i.e. -hypersurface) which is diffeomorphic to a sphere. S. Brendle confirmed the conjecture for 2-dimensional -hypersurfaces. For any dimensional -hypersurfaces, if , we constructed compact convex embedded -hypersurface which is diffeomorphic to a sphere and is not a round sphere. In this paper, for , we construct a compact mean convex embedded -hypersurface which is diffeomorphic to a sphere and is not a round sphere. In fact, for , there are no compact convex embedded -hypersurfaces which are diffeomorphic to spheres except a round sphere.

Paper Structure

This paper contains 7 sections, 29 theorems, 68 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Behavior of the solutions to (\ref{['0526eq2.8']}) and (\ref{['231212eq2.9']})
  4. Global splicing and classification of profile curves
  5. Behavior of the curve $\bar{\gamma}_b$
  6. Behavior of the curve $\gamma_\delta$
  7. Proof of the theorem

Key Result

Theorem 1.1

For $n \ge 2$ and $-\frac{2}{\sqrt{n+2}} < \lambda <0$, there exists an embedded convex $\lambda$-hypersurface $\Sigma^n \subset \mathbb{R}^{n+1}$ which is diffeomorphic to $\mathbb{S}^n$ and is not isometric to a standard sphere.

Figures (5)

  • Figure 5.1: Graphs of some $\overline{\gamma}_b$ with $-R_\lambda < b < -\lambda$
  • Figure 6.1: Graphs of some ${\gamma}_\delta$ when $\delta$ is close to 0
  • Figure 6.2: Graphs of some ${\gamma}_\delta$ when $\delta$ is close to $C_\lambda$
  • Figure 7.1: A curve which generates a mean convex $\mathbb S^n$$\lambda$-hypersurface
  • Figure 7.2: A mean convex $\lambda$-sphere in $\mathbb{R}^3$ where $\lambda=\sqrt 5$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 42 more