Minimal Surfaces and Gauss Curvature of Conoid in Finsler Spaces with ( α , β )-Metrics *

In this paper, minimal submanifolds in Finsler spaces with (α, β)-metrics are studied. Especially, helicoids are also minimal in (α, β)-Minkowski spaces. Then the minimal surfaces of conoid in Finsler spaces with (α, β)-metrics are given. Last, the Gauss curvature of the conoid in the 3-dimension Randers-Minkowski space is studied.


Introduction
In recent decades, geometry of submanifolds in Finsler geometry has been rapidly developed.By using the Busemann-Hausdorff volume form, Z. Shen [1] introduced the notions of mean curvature and normal curvature for Finsler submanifolds.Being based on it, Bernstein type theorem of minimal rotated surfaces in Randers-Minkowski space was considered in [2].Later, Q.He and Y. B. Shen used another important volume form, i.e., Holmes-Thompson volume form, to introduce notions of another mean curvature and the second fundamental form [3]. Thus, Q.He and Y. B. Shen constructed the corresponding Bernstein type theorem in a general Minkowski space [4].
The theory of minimal surfaces in Euclidean space has developed into a rich branch of differential geometry.A lot of minimal surfaces have been found in Euclidean space.Minkowski space is an analogue of Euclidean space in Finsler geometry.A natural problem is to study minimal surfaces with Busemann-Hausdorff or Holmes-Thompson volume forms.M. Souza and K. Tenenblat first studied the minimal surfaces of rotation in Randers-Minkowski spaces, and used an ODE to characterize the BH-minimal rotated surfaces in [5].Later, the nontrivial HT-minimal rotated hypersurfaces in quadratic (α, β)-Minkowski space are studied [6].N. Cui and Y. B. Shen used another method to give minimal rotational hypersurface in quadratic Minkowski (α, β)-space [7].However, these examples only consider the special (α, β)metrics either Randers or quadratic.Therefore, what is the case with the general (α, β)-metric?
The main purpose of this paper is to study the conoid in (α, β)-space.It includes minimal submanifolds in  Finsler spaces with general (α, β)-metric and the Causs curvature in Randers-Minkowski 3-space.
We present the equations that characterize the minimal hypersurfaces in general (α, β)-Minkowski space.We prove that the conoid in Minkowski 3-space with metric is minimal if and only if it is a helicoid or a plane under some conditions.Finally, similar to [7], we give the Gauss curvature of conoid in Randers-Minkowski 3-space and point out that the Gauss curvature is not always nonpositive on minimal surfaces.

Preliminaries
Let M be an n-dimensional smooth manifold.A Finsler metric on M is a function satisfying the following properties: 1) F is smooth on for all   ; 3) The induced quadratic form g is positively definite, where Here and from now on, , and we shall use the following convention of index ranges unless otherwise stated: The projection gives rise to the pullback bundle and its dual , which sits over .We shall work on and rigidly use only objects that are invariant under positive rescaling in y, so that one may view them as objects on the projective sphere bundle SM using homogeneous coordinates.
, called the Hilbert form, whose dual is , called the distinguished field.The volume element SM of SM with respect to the Riemannian metric g , the pull-back of the Sasaki metric on , can be expressed as where   The volume form of a Finsler n-manifold (M, F) is defined by where denotes the volume of the unit Euclidean (n − 1)-sphere , Let (M, F) and  M F   be Finsler manifolds, and : for the isometric immersion with respect to g  , and set 2 , , , where where which is called the mean curvature form of f.An isometric immersion is the critical point of its volume functional with respect to any variation vector field.Then f is minimal if and only if   .
It have been proved where In the following part, we will discuss minimal hypersurfaces in Minkowski space with (α, β)-metric.
  be the unit norm eld of al vector fi    it n mal vector fiel ect to be the un or d of M with resp g  ely.That is , respectiv 0, 0, From above, we know that f is mini al if and only if 3) and (3.4), a  nd in a similar way as in [5], we n get is a sphere such that 1 where   .Now, we consider the conoid in 3-dimensional (α, β)-Minkowski space paralleling to x 3 -axis.Set where 1.
Note that al vector of the surface is  w The , , b b b    are not all zeros.To simplify the computation, we only discuss quadratic (α, β)-metric: where . Then (3.8) becomes an equation respect to u: That is to say a minimal conoid hypersurface is a plane with respect to the given metric above.
are not all zeros).Then a m ne.

Gauss Curvature of Conoid in Randers
all kn m dy the Gauss curvatu n Minkowski-Randers 3-space around x 3 -axis in the direction

3-Space
As we own, the Gauss curvature of a minimal surface is nonpositive everywhere in Euclidean space.Then, a fact ho natural problem arises: whether this lds for inimal surfaces in Minkowski-Randers 3-space?In this section, we stu re of conoid i . Then te Deno ature is computed in Euclidean space as follows: . h . By a direct computation, we have 2) and (4.3), we obtain the following theorem.

Theorem 4 Let
Then, from   3 , V F  be an Randers-Minkow space with  , then       , 0 K x y  .In sum, the Gauss curvature is not nonpositive anywhere.

h uh uh u h u h u h u h h uh h u u h u h u h u h h u h u h u h
 on its tangent bundle.In this section we shall use the convention that 1 used.Note that the induced 1-form * f     on the surface is closed.Then the Ricci curvature tensor of * F f F   is given by ([10], Page 118)

3 2
Note that a helicoid is minimal if and only if it is a conoid with respect β)-metrics (where by