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

作者：Cui, Ningwei^*; Zhou, Linfeng

来源：Tohoku Mathematical Journal, 2022, 74(4): 627-656.

DOI：10.2748/tmj.20210907a

• Finsler geometry
• general (alpha, beta)-metric
• mean curvature
• Randers manifold
• minimal surface

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.

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  西南交通大学