Tohoku Mathematical Journal
2022
December
SECOND SERIES VOL. 74, NO. 4

Tohoku Math. J.
74 (2022), 627-656

Title THE VARIATION OF A FUNCTIONAL ON THE RIEMANNIAN SUBMANIFOLDS AND ITS APPLICATIONS

Author Ningwei Cui and Linfeng Zhou

(Received June 1, 2021, revised September 7, 2021)
Abstract. The aim of this paper is to derive the first variation of a functional on an oriented submanifold in the Riemannian manifold involving an arbitrary vector field. After obtaining the first variation formula, a notion of $\sigma$-mean curvature is naturally introduced. We also find an important identity involving the $\sigma$-mean curvature and the divergence of the tangent component of the vector field.
 As an application, we get a general formula of the $\sigma$-mean curvature for the hypersurfaces in a class of Finsler manifolds called the general $(\alpha,\beta)$-manifolds (introduced in [38]). Hence the vanishing $\sigma$-mean curvature characterizes the minimal hypersurfaces under the Busemann-Hausdorff measure ([29]) and the Holmes-Thompson measure ([23]). We also give a general formula for the hypersurface in a Randers manifold involving the navigation data without any restriction on the vector field.
 In terms of the identity, we prove some nonexistence theorems of the closed orientable minimal submanifold in some special non-Minkowskian Finsler manifolds.

Mathematics Subject Classification. Primary 53C40; Secondary 53C60.
Key words and phrases. Finsler geometry, general $(\alpha,\beta)$-metric, mean curvature, Randers manifold, minimal surface.

To the top of this page

Back to the Contents