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Delaunay hypersurfaces in spheres

Yongsheng Zhang

Abstract

We study Delaunay hypersurfaces in $\mathbb S^n$ with $n\geq 3$ and add a missing (flower) type of the category. Moreover, embedded Delaunay hypersurfaces of nonzero constant mean curvatures in $\mathbb S^n$ are found.

Delaunay hypersurfaces in spheres

Abstract

We study Delaunay hypersurfaces in with and add a missing (flower) type of the category. Moreover, embedded Delaunay hypersurfaces of nonzero constant mean curvatures in are found.
Paper Structure (8 sections, 21 theorems, 35 equations, 10 figures)

This paper contains 8 sections, 21 theorems, 35 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The $h$-CMC equation
  3. Local construction of $h$-CMC Delaunay hypersurfaces
  4. Global construction of $h$-CMC Delaunay hypersurfaces
  5. Case of positive $h$.
  6. Case of vanishing $h$.
  7. Case of negative $h$.
  8. Descendent to (immersed or embedded) closed CMC hypersurfaces

Key Result

Proposition 3.1

Assume that $h>0$. If $C\in [0, C_h)$, then the largest possible open domain $I(C, h)$ is decided by two intersection points of closures of graphs of $R(\cdot\,, C, h)$ and $L(\cdot)$. When $C\in (-\frac{h}{n-1},0)$, distinctly $I(C, h)$ is determined by intersection points of the graph of $R(\cdot\

Figures (10)

  • Figure I: Delaunay $h$-CMC hypersurfaces with $h>0$
  • Figure II: Graphs of $R(s, C, \pm h)$ for $h>0$ and contact points
  • Figure III: Lagest domain for $\Theta>0$ in \ref{['n3.5']} when $h>0$
  • Figure IV: Reflection extension when $h>0$ and $C\in (0, C_h)$
  • Figure V: Reflection extension for $C=0$ with $h>0$
  • ...and 5 more figures

Theorems & Definitions (47)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • Remark 4.5
  • ...and 37 more