Hypersurfaces with Constant Mean Curvatures on Finsler manifolds

TL;DR

This paper explores hypersurfaces with constant mean curvature in Finsler manifolds, establishing geometric interpretations, relationships between principal curvatures, and deriving inequalities and theorems analogous to classical Riemannian results.

Contribution

It introduces a geometric meaning for such hypersurfaces using volume preserving variations and links principal curvatures through homothetic navigation in Finsler spaces.

Findings

Established a geometric interpretation of constant mean curvature hypersurfaces.

Derived a Heintze-Karcher type inequality in Finsler spaces.

Proved an Alexandrov type theorem in specific Finsler geometries.

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.