On minimal surfaces in a class of Finsler 3-spheres

Perturbing the classical metric on the round 3-sphere by the Killing vector fields tangent to Hopf fibers, one gets a class of Finsler metrics of Randers type with constant flag curvature , depending on one parameter , 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 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.