On minimal surfaces in a class of Finsler 3-spheres

Abstract

Perturbing the classical metric on the round 3-sphere \(S^3\) by the Killing vector fields tangent to Hopf fibers, one gets a class of Finsler metrics of Randers type with constant flag curvature \(\mathbf{K}=1\), depending on one parameter \(k>1\), called Bao–Shen’s (J Lond Math Soc 66:453–467, 2002) metrics. The corresponding spheres will be called Bao–Shen’s spheres, which are proper candidates of positively curved Finsler space forms. In this paper, we study the minimal surfaces in Bao–Shen’s spheres. We first study submanifolds isometrically immersed in a Randers manifold by the method of Zermelo’s navigation. Then we give a clear formula of the mean curvature of the surface in a Bao–Shen’s sphere by introducing the volume ratio function to show its relation with the mean curvature of the surface in round 3-sphere. As an application, we find an interesting family of minimal surfaces with respect to Busemann–Hausdorff volume form in Bao–Shen’s sphere called helicoids. This family contains the compact minimal surfaces \(\tau _{m,n}\) in round 3-sphere constructed by Lawson (Ann Math 92(3):335–374, 1970), including great 2-spheres, Clifford torus, Klein bottles, etc. Moreover, two rigidity results are given.