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Geometry
Volume 2013 (2013), Article ID 718272, 7 pages
http://dx.doi.org/10.1155/2013/718272
Research Article

Hypersurfaces with Null Higher Order Anisotropic Mean Curvature

1School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China
2Research Institute of Mathematics and Applied Mathematics, Shanxi University, Taiyuan 030006, China

Received 18 April 2013; Accepted 11 June 2013

Academic Editor: Reza Saadati

Copyright © 2013 Hua Wang and Yijun He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given a positive function on which satisfies a convexity condition, for , we define for hypersurfaces in the th anisotropic mean curvature function , a generalization of the usual th mean curvature function. We call a hypersurface anisotropic minimal if , and anisotropic -minimal if . Let be the set of points which are omitted by the hyperplanes tangent to . We will prove that if an oriented hypersurface is anisotropic minimal, and the set is open and nonempty, then is a part of a hyperplane of . We also prove that if an oriented hypersurface is anisotropic -minimal and its th anisotropic mean curvature is nonzero everywhere, and the set is open and nonempty, then has anisotropic relative nullity .

1. Introduction

Let be a smooth function which satisfies the following convexity condition: where is the standard unit sphere in , denotes the intrinsic Hessian of on , denotes the identity on , and >0 means that the matrix is positive definite. We consider the map its image is a smooth, convex hypersurface in called the Wulff shape of (see [19]). When , the Wulff shape is just .

Now let be a smooth immersion of an oriented hypersurface. Let denote its Gauss map. The map is called the anisotropic Gauss map of .

Let . is called the -Weingarten operator, and the eigenvalues of are called anisotropic principal curvatures. Let be the elementary symmetric functions of the anisotropic principal curvatures : We set . The th anisotropic mean curvature is defined by , also see Reilly [10]. is called the anisotropic mean curvature. When , is just the Weingarten operator of hypersurfaces, and is just the th mean curvature of hypersurfaces which has been studied by many authors (see [1114]). Thus, the th anisotropic mean curvature generalizes the th mean curvature of hypersurfaces in the -dimensional Euclidean space .

We say that is anisotropic -minimal if .

For , we define . We call the anisotropic relative nullity; it generalized the usual relative nullity.

For a smooth immersion of a hypersurface into an -dimensional space form with constant sectional curvature , we denote by where for every , is the totally geodesic hypersurface of tangent to at . So, in the case of , is the set of points which are omitted by the hyperplanes tangent to .

We will study immersion with nonempty. In this direction, Hasanis and Koutroufiotis (see [15]) proved the following.

Theorem 1. Let be a complete minimal immersion with . If is nonempty, then is totally geodesic.

Later, in [16], Alencar and Frensel extended the result above assuming an extra condition. They proved the following.

Theorem 2. Let be an oriented, minimally immersed hypersurface. If   is open and nonempty, then is totally geodesic.

In [17], Alencar and Batista studied hypersurfaces with null higher order mean curvature; they proved the following.

Theorem 3. Let be a complete and orientable Riemannian manifold and let be an isometric immersion with and everywhere, . If   is open and nonempty, then the relative nullity .

We note that, Alencar in [18] provides examples of nontotally geodesic minimal hypersurfaces in , , with nonempty ; in [17], Alencar and Batista provides examples of 1-minimal hypersurfaces with everywhere in ,  , with nonempty but . These examples show that it is necessary to add an extra hypothesis.

In this paper, we prove the anisotropic version of Theorems 2 and 3 for an immersion . Explicitly, we prove the following two theorems.

Theorem 4. Let be an oriented, anisotropic minimally immersed hypersurface. If   is open and nonempty, then is a part of a hyperplane of .

Theorem 5. Let be an oriented immersed hypersurface with and everywhere, . If is open and nonempty, then the anisotropic relative nullity .

2. Preliminaries

In this paper, we use the summation convention of Einstein and the following convention of index ranges unless otherwise stated:

We define to be then is a Minkowski norm on . In fact, as proved in [19], is smooth and we have the following.

Proposition 6. (1)  , for all ;
(2)  , for all , ;
(3)  , for all , and the equality holds if and only if , or or for some .
(4)  .

We define where and , .

From the Euler’s theorem for homogeneous functions, we have where . Thus, where is nonzero everywhere and .

As is a Minkowski norm on , the following lemma holds (see [20, 21]).

Lemma 7. For any and one has and the equality holds if and only if there exists such that .

Let be an oriented hypersurface in the Euclidean space . Let denote its anisotropic Gauss map. Then for any , is perpendicular to with respect to the inner product and . Thus, we call an anisotropic unit normal vector of .

3. A Connection on Hypersurfaces of Minkowski Space

Let be an oriented hypersurface in the Euclidean space and denote its anisotropic Gauss map.

Let be the standard connection on the -dimensional Euclidean space . For vector fields on , we decompose as the tangent part and the anisotropic normal part II with respect to the inner product . That is, where .

We also have the Weingarten formula: where we have used (9).

It is easy to verify that is a torsion free connection on and is a symmetric second order covariant tensor field on . We call the anisotropic second fundamental form.

Let be a local frame of and its dual frame. Let , , , , where is the inverse matrix of . Then we have

Differentiating (13) and using (14), we get

Differentiating (14) and using (14)-(15), we get where and .

Differentiating (15) and using (14), we get where

Note is the matrix of the -Weingarten operator , its eigenvalues are called the anisotropic principal curvatures, and we denote them by .

We have invariants, the elementary symmetric function of the anisotropic principal curvatures: For convenience, we set . The th anisotropic mean curvature is defined by

Using the characteristic polynomial of , is defined by

So, we have where is the usual generalized Kronecker symbol; that is, equals +1 (resp., −1) if are distinct and is an even (resp., odd) permutation of and in other cases it equals zero.

Definition 8. Let be a smooth function. One defines the gradient (with respect to the induced metric on ) of the function by where is any smooth vector field on .
Define by ; then

We define where is the determinant of the matrix . Then is a volume element on .

Definition 9. Let be a smooth vector field on . One defines the divergence (with respect to the volume element ) by , where

Lemma 10. Let ; then , where

Proof. By (14), (15), we get From the definition of , we have So,

4. Operator for Hypersurfaces

We introduce the Newton transformation defined by then

Lemma 11. The matrix of is given by:

Proof. We prove Lemma 11 inductively. For , it is easy to check that (35) is true.
We can check directly
Assume that (35) is true for , we only need to show that it is also true for . For , using (24) and (36), we have

Lemma 12. For each , one has(a) ;(b) ;(c) ;(d) .

Proof. (a) Noting is skew symmetric in and is symmetric in (from (19), we have (b) Using (35) and (24), we have (c) Using (b) and the definition of , we have (d) Using (b) and the definition of , we have

Remark 13. When , Lemma 12 was a well-known result (e.g., see Barbosa and Colares [22], or Reilly [23]).

Lemma 14. One has

Proof. From the definition of , we have the following calculation:

We define an operator by

In the sequel, we will need the following lemma. Item (a) is essentially the content of Lemma and Equation (1.3) in [24], while item (b) is quoted as Proposition  1.5 in [25].

Lemma 15. Let be an oriented hypersurface, and , .(a)If , then is semidefinite at ;(b)if and , then is definite at .

Another important result is as follows (see [26]).

Lemma 16. Let be an oriented hypersurface, and .(a)For , one has . Moreover, if equality happens for or for some , with in this case, then is an anisotropic umbilical point (i.e. );(b)if, for some , one has , then for all . In particular, at most of the anisotropic principal curvatures are different from zero.

The result below is standard, so we omit the proof.

Lemma 17. Let be an oriented hypersurface. The operator associated to the immersion is elliptic if and only if is positive definite.

Definition 18. Let be a smooth function. The Laplacian is defined by .

It is easy to see that is an elliptic differential operator.

Definition 19. Let be an immersed hypersurface and its anisotropic unit normal vector field. The function is called the support function of the immersion .

Next, we compute for the support function .

Differentiating the decomposition we obtain So, from (13), (14), and (15) we have Thus, we get Denote , , by respectively. Then we have (using (19) the following By using Lemmas 12 and 14, we get Thus, we proved the following lemma.

Lemma 20. For , one has the following

Remark 21. Recall and ; let in (52); we get

5. Proof of Theorems 4 and 5

We fix a point as the origin of . Without loss of generality, we assume, for each , is the anisotropic unit normal vector of at such that (otherwise we consider the function instead). This gives an orientation to ; indeed, the component of the position vector perpendicular (with respect to the inner product ) to defines a never zero, anisotropic normal, vector field on , such that the support function is positive on .

5.1. Proof of Theorem 4

Since is anisotropic minimal, from (53) we get Let . We claim that is attained at some point . Consider a sequence such that as . To each we associate ; then . Since is bounded, there exists a subsequence, which again we call , such that for some . Since is closed and we deduce that for some . Thus, by the continuity of and Lemma 7, so as needed. Now, from the usual maximum principle is constant, . From (54) we then have and is totally geodesic.

5.2. Proof of Theorem 5

Since , from Lemma 20 we get Using Lemma 15(a) we have that is semidefinite. Since does not vanish, we have that is positive or negative, because , where . Now we use Lemma 16 and obtain the following: Using the information above, we claim that .

Case  (i) . In this case, is positive definite, and is elliptic by Lemma 17. Using (57) we conclude that . Whereas from (56) we have Following exactly the proof as in Theorem 4, we conclude that is constant, . From (56) we then have .

Case  (ii) . In this case, is negative definite, and is elliptic by Lemma 17. Using (57) we conclude that . Whereas from (56) we have

Now, following exactly the proof as in Theorem 4, we conclude that is constant, . From (56) we then have .

Thus we conclude that . Now, we use Lemma 16(b) to conclude that for and so that . Since does not change sign we have that .

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