Minimal Surfaces In General(α,β)-spaces

Finsler geometry is an important branch of differential geometry.Under the initiative of Professor Shiing Shen Chem,Finsler geometry has been made a great progress after 1990s.In recent years,more and more researchers pay attention to Finsler submanifolds which becomes a research focus in Finsler geometry.In this paper we study the structures and properties of minimal surfaces in some three and four-dimensional Finsler spaces.Firstly,we study surfaces produced by a one-parameter subgroup acting on a curve in some 4-dimensional non-Minkowski general(α,β)-space(M4,F).We completely characterize the minimality of these surfaces with particular Finsler metrics.We also study the asymptotic be-haviour of one kind of these minimal surfaces.Moreover,we construct a kind of ruled hyper-surfaces in(M4,F)and study the minimality of them.Secondly,we study the minimal surfaces in a 3-dimensional Randers space(V3,F).We obtain the explicit parameterized solutions of the equation given by Souza and Tenenblat in[1].We discuss the convexity and the asymptotic property of the meridian curves of the minimal rota-tional surfaces.By studying the existence,we prove that planes are the only minimal translation surfaces in(V3,F).Finally,we study a class of Randers metrics F = α+β depended on the navigation data(H,v),where H is the standard Riemannian metric on S2 and v =(kx + ly,-lx + ky)is an arbitrary vector field.We obtain the sectional curvature of α and the S-curvature of F.Moreover,we proof that whether the S-curvature is isotropic depends on k.