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Prescribed mean curvature hypersurfaces in conformal product manifolds

Qiang Gao, Hengyu Zhou

Abstract

In this paper we give the existence of prescribed mean curvature (PMC) hypersurfaces in conformal product manifolds with (possibly empty) $C^{1,α}$ fixed graphical boundaries under a barrier condition. This generalizes Gerhardt's result in conformally flat spaces. It provides new examples of the Plateau problem of PMC hypersurfaces with clear topology under high dimensions. In addition, if a quasi-decreasing condition of PMC functions is satisfied, such PMC hypersurfaces are $C^1$ graphs.

Prescribed mean curvature hypersurfaces in conformal product manifolds

Abstract

In this paper we give the existence of prescribed mean curvature (PMC) hypersurfaces in conformal product manifolds with (possibly empty) fixed graphical boundaries under a barrier condition. This generalizes Gerhardt's result in conformally flat spaces. It provides new examples of the Plateau problem of PMC hypersurfaces with clear topology under high dimensions. In addition, if a quasi-decreasing condition of PMC functions is satisfied, such PMC hypersurfaces are graphs.
Paper Structure (9 sections, 19 theorems, 91 equations)

This paper contains 9 sections, 19 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. conformal product manifolds and $\Lambda$-perimeter minimizers
  3. Conformal product manifolds
  4. Some facts on $\Lambda$-perimeter minimizers
  5. Limit of bounded graphs with bounded mean curvature
  6. Existence of PMC hypersurfaces
  7. Quasi-decreasing case
  8. Quasi-decreasing condition
  9. An application

Key Result

Theorem 1.3

Suppose $(gr(u_1), gr(u_0))$ is a barrier of $(M_f, \mathcal{H})$ and $2\leq n\leq 7$. Then there is a $C^{3,\alpha}$ embedded hypersurface $\Sigma$ with a fixed graphical boundary $gr(\psi)$ (possibly empty) such that $\Sigma \cup gr(\psi)$ is homeomorphic to $N$ and its mean curvature is $\mathcal

Theorems & Definitions (50)

  • Definition 1.2
  • Theorem 1.3: Theorem \ref{['thm:main']}
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Definition 2.6
  • ...and 40 more