Taiwanese Journal of Mathematics

Hypersurfaces of Randers Spaces with Constant Mean Curvature

Jintang Li

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Abstract

Let $(\overline{M}^{n+1}, \overline{F})$ be a complete simply connected Randers space with $\overline{F}(x,Y) = \overline{a}(x,Y)+ \overline{b}(x,Y)$, where $\overline{a}(x,Y) = \sqrt{\overline{a}_{ij}(x) Y^i Y^j}$ and $\overline{b}(x,Y) = \overline{b}_i(x) Y^i$ are a Riemannian metric and a $1$-form on the smooth $(n+1)$-dimensional manifold $\overline{M}$ respectively. Assume the $1$-form $\overline{b}$ is parallel with respect to $\overline{a}$ and the sectional curvature $\overline{K}_{\overline{M}}$ of $\overline{M}$ with respect to $\overline{a}$ satisfies $\delta(n) \leq \overline{K}_{\overline{M}} \leq 1$. In this paper, we study the compact hypersurface $(M,F)$ of the Randers space $(\overline{M}^{n+1}, \overline{F})$ with constant mean curvature $|H|$ and prove that if the norm square $S$ of the second fundamental form of $(M,F)$ with respect to the Finsler metric $\overline{F}$ satisfies a certain inequality, then $S = n|H|^2$ and $M$ is the unit sphere or equality holds. In that case, we describe all $M$ that satisfy this equality, which generalizes the result of [8] from the Riemannian case to the Randers space.

Article information

Source
Taiwanese J. Math., Volume 21, Number 5 (2017), 979-996.

Dates
Received: 24 August 2016
Revised: 27 November 2016
Accepted: 3 January 2017
First available in Project Euclid: 1 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1501599179

Digital Object Identifier
doi:10.11650/tjm/7945

Mathematical Reviews number (MathSciNet)
MR3707880

Zentralblatt MATH identifier
06871355

Subjects
Primary: 53C60: Finsler spaces and generalizations (areal metrics) [See also 58B20]
Secondary: 53C40: Global submanifolds [See also 53B25]

Keywords
Finsler manifolds Randers spaces hypersurfaces

Citation

Li, Jintang. Hypersurfaces of Randers Spaces with Constant Mean Curvature. Taiwanese J. Math. 21 (2017), no. 5, 979--996. doi:10.11650/tjm/7945. https://projecteuclid.org/euclid.twjm/1501599179


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