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On $O(p)\times O(q)$-invariant constant mean curvature hypersurfaces with singularity

Hilário Alencar, Ronaldo Garcia, Gregório Silva Neto

Abstract

We classify the $O(p)\times O(q)$-invariant constant mean curvature hypersurfaces with singularity at the origin, solving a conjecture of Wu-yi Hsiang.

On $O(p)\times O(q)$-invariant constant mean curvature hypersurfaces with singularity

Abstract

We classify the -invariant constant mean curvature hypersurfaces with singularity at the origin, solving a conjecture of Wu-yi Hsiang.
Paper Structure (3 sections, 8 theorems, 61 equations, 7 figures)

This paper contains 3 sections, 8 theorems, 61 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main-1']}
  3. Proof of Theorem \ref{['thm:main-2']}

Key Result

Theorem 1.1

There is only one hypersurface of $\, \mathbb{R}^{p+q}$ invariant by $O(p)\times O(q),$ with constant mean curvature $H\ne 0$, whose generating curve is a global solution of eq:mean-H0 with one cusp point at the origin. Moreover,

Figures (7)

  • Figure 1.1: Singular extended solutions asymptotic to the coordinate axes.
  • Figure 1.2: Sketch of the generating curve of the solution with one cusp point (singular point) at the origin.
  • Figure 2.1: Integral curves of $\tilde{Y}_0$ in the region $[0,\pi/2]\times [0,2\pi]$. Left: The singular point $p_5$ (resp. $p_6$) is an attractor (resp. repeller). Here $p+q<8$. Both are foci. Right: The singular point $p_5$ (resp. $p_6$) is an attractor (resp. repeller). Here $p+q\geq 8$. Both are nodes.
  • Figure 2.2: Blowing-up and the local invariant manifolds $W^u(P_5)$ and $W^s(P_6)$.
  • Figure 2.3: Projections of the invariant manifolds $W^u(P_5)$ and $W^s(P_6)$.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Conjecture
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 9 more