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Characterizations of umbilical hypersurfaces by partially overdetermined problems in space forms

Yangsen Xie

Abstract

In this paper, we characterize the rigidity of umbilical hypersurfaces by a Serrin-type partially overdetermined problem in space forms, which generalizes the similar results in Euclidean half-space and Euclidean half-ball. Guo-Xia first obtained these rigidity results when the Robin boundary condition on the support hypersurface is homogeneous, at this time the target umbilical hypersurface has orthogonal contact angle with the support. However, in this paper we can obtain any contact angle $θ\in (0,π)$ by changing the Robin boundary condition to be inhomogeneous.

Characterizations of umbilical hypersurfaces by partially overdetermined problems in space forms

Abstract

In this paper, we characterize the rigidity of umbilical hypersurfaces by a Serrin-type partially overdetermined problem in space forms, which generalizes the similar results in Euclidean half-space and Euclidean half-ball. Guo-Xia first obtained these rigidity results when the Robin boundary condition on the support hypersurface is homogeneous, at this time the target umbilical hypersurface has orthogonal contact angle with the support. However, in this paper we can obtain any contact angle by changing the Robin boundary condition to be inhomogeneous.
Paper Structure (6 sections, 13 theorems, 111 equations, 7 figures)

This paper contains 6 sections, 13 theorems, 111 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of auxiliary functions and some integral formulas
  4. Existence of auxiliary functions
  5. Some integral formulas
  6. Integral identities and proof of Theorem \ref{['Theorem']}

Key Result

Theorem 1.1

Assume the partially overdetermined BVP Guo admits a weak solution $u\in W_0^{1,2}(\Omega,\Sigma)$, i.e. together with an additional boundary condition $\partial_\nu u=c$ on $\Sigma$. Assume further that $u\in W^{1,\infty}(\Omega)\cap W^{2,2}(\Omega)$. Then $c>0$ and $\Sigma$ must be part of an umbilical hypersurface with principal curvature $\frac{1}{(n+1)c}$ which intersects $S_{K,\kappa}$ orth

Figures (7)

  • Figure 2.1:
  • Figure 2.2: $S_{K,\kappa}$ is a geodesic sphere with $\kappa=\coth R>1$, and the shaded area is $B_{K,\kappa}^{\rm{int},+}$.
  • Figure 2.3: $S_{K,\kappa}$ is an equidistant hypersurface with $\kappa=\cos\alpha\in(0,1]$, and the shaded area is $B_{K,\kappa}^{\rm int}$.
  • Figure 2.4: $S_{K,\kappa}$ is a totally geodesic hyperplane with $\kappa=0$, and the shaded area is $B_{K,\kappa}^{\rm int}$.
  • Figure 2.5: $S_{K,\kappa}$ is a geodesic sphere with $\kappa=\cot R\in[0,\infty)$, and the shaded area is $B_{K,\kappa}^{\rm{int},+}$.
  • ...and 2 more figures

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • Proposition 3.1
  • ...and 14 more