Nontrivial minimal surfaces in a class of Finsler spheres of vanishing [formula omitted]-curvature

Abstract

The minimal surface theory in Finsler geometry deserves to be well developed as that in Riemannian geometry. In this paper, we derive the mean curvature of submanifolds in a general (α,β)-manifold by considering the Busemann–Hausdorff measure and Holmes–Thompson measure respectively. We then study the rotationally invariant minimal surfaces, in the Finsler 3-sphere endowed with an (α,β)-metric F̃k=α̃kϕ(β̃k/α̃k), k>1, where ϕ is a smooth function, (S3,α̃k) is the Berger sphere E(4/k,1) and β̃k is a Killing one form of constant length along the Hopf fibers of S3. We define the energy of the minimal surfaces, and by using the volume ratio function introduced by the author and Y.-B. Shen, we give the explicit local expressions of the rotationally invariant BH-minimal and HT-minimal surfaces in such sphere, respectively. As a special case, we give a detailed study of the rotationally invariant HT-minimal surfaces in the 3-sphere with square metric.