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Hypersurfaces with Constant Mean Curvatures on Finsler manifolds

Yali Chen, Qun He, Yantong Qian

Abstract

In this paper, we give the geometric meaning of hypersurfaces with constant mean curvature in a Finsler manifold by using volume preserving variation. Then we give the correspondence between principal curvatures of submanifolds by a homothetic navigation, which means that some geometric properties of submanifolds are the same. Finally, we deduce a Heintze-Karcher type inequality and prove an Alexandrov type theorem in special Finsler spaces.

Hypersurfaces with Constant Mean Curvatures on Finsler manifolds

Abstract

In this paper, we give the geometric meaning of hypersurfaces with constant mean curvature in a Finsler manifold by using volume preserving variation. Then we give the correspondence between principal curvatures of submanifolds by a homothetic navigation, which means that some geometric properties of submanifolds are the same. Finally, we deduce a Heintze-Karcher type inequality and prove an Alexandrov type theorem in special Finsler spaces.
Paper Structure (14 sections, 22 theorems, 84 equations)

This paper contains 14 sections, 22 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finsler manifolds
  4. Immersed Submanifolds
  5. Variation of the induced volume
  6. Volume preserving variation
  7. The definition in Finsler geometry
  8. Proof of Theorem \ref{['thm1_1']}
  9. Geometric correspondence of submanifolds by homothetic navigation
  10. Heintze-Karcher Type Inequality
  11. Anisotropic mean curvature in Euclidean spaces
  12. Heintze-Karcher Type Inequality in Minkowski spaces
  13. Proof of Theorem \ref{['thm1_3']}
  14. Proof of Theorem \ref{['thm1_4']}

Key Result

Theorem 1.1

Let $\phi : M\rightarrow N$ be an embedded hypersurface in a Finsler manifold $(N,F)$. Then $\phi$ is a critical point of any volume preserving variation if and only if the mean curvature of $M$ is a non-zero constant in $N$ with respect to any induced volume.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 3.1
  • Definition 3.2
  • ...and 17 more