Hypersurfaces of Randers spaces with constant mean curvature
Let \((\overline{M}^{n+1},\overline{F})\) be a complete simply connected smooth \((n+1)\)-dimensional Randers space with \(\overline{F}(x,y)=\overline{a}(x,y)+\overline{b}(x,y)\), where \(\overline{a}(x,y)=\sqrt{\overline{a}_{ij}(x)y^iy^j}\) and \(\overline{b}(x,y)=\overline{b}_i(x)y^i\) are a Riemannian metric and a 1-form on the manifold \(\overline{M}\), respectively. In this paper the author studies the hypersurfaces of Randers space \((\overline{M}^{n+1},\overline{a}+\overline{b})\) with \(\overline{b}\) parallel with respect to \(\overline{a}\) and obtain the following classification theorem: Assume \(\overline{M}^{n+1}\) is a \(\delta(n)\)-pinching Riemannian manifold with respect to the Riemannian metric \(\overline{a}\), i.e., the sectional curvature \(\overline{K}_{\overline{M}}\) of \(\overline{M}\) with respect to \(\overline{a}\) satisfies \(\delta(n)\leq \overline{K}_{\overline{M}}\leq 1\). If \(M^n\) is a compact hypersurface of \((\overline{M}^{n+1},\overline{F})\) with constant mean curvature \(|H|\) and the norm square \(S\) of the second fundamental form of \((M,F)\) with respect to the Finsler metric \(\overline{F}\) satisfies \(\alpha(1-\delta)\leq S-n{|H|}^2\leq B_H\), where \(\alpha\) is a certain constant depending on \(n\) and \(B_H\) is the positive solution of a certain algebraic equation, then either \(M\) is the unit sphere or \(S-n{|H|}^2=B_H\) and one of the following cases occurs: (1) \(H=0\) and \(M=S^k(\sqrt {k/n})\times S^{n-k}(\sqrt {(n-k)/n})\) is a minimal Clifford hypersurface, (2) \(H\neq 0, n\geq 3\) and \(M=S^1(r_1)\times S^{n-1}(r_2)\) is an \(H\)-torus, where \({r_1}^2+{r_2}^2=1\) and \({r_2}^2<(n-1)/2\), (3) \(H\neq 0, n=2\) and \(M=S^1(r_1)\times S^1(r_2)\) is an \(H\)-torus, where \({r_1}^2+{r_2}^2=1\) and \({r_1}^2\neq 1/2\). This theorem generalizes the result of \textit{H.-W. Xu} [Trans. Am. Math. Soc. 347, No. 5, 1743--1751 (1995; Zbl 0830.53048)] from the Riemannian case to the Randers case.
scientific article; zbMATH DE number 6871355