STABILITY OF GENERALIZED LINEAR WEINGARTEN HYPERSURFACES IMMERSED IN THE EUCLIDEAN SPACE

Given a positive function F defined on the unit Euclidean sphere and satisfying a suitable convexity condition, we consider, for hypersurfaces Mn immersed in the Euclidean space Rn+1, the so-called k-th anisotropic mean curvatures HF k , 0 ≤ k ≤ n. For fixed 0 ≤ r ≤ s ≤ n, a hypersurface Mn of Rn+1 is said to be (r, s, F )-linear Weingarten when its k-th anisotropic mean curvatures HF k , r ≤ k ≤ s, are linearly related. In this setting, we establish the concept of stability concerning closed (r, s, F )-linear Weingarten hypersurfaces immersed in Rn+1 and, afterwards, we prove that such a hypersurface is stable if, and only if, up to translations and homotheties, it is the Wulff shape of F . For r = s and F ≡ 1, our results amount to the standard stability studied, for instance, by Alencar–do Carmo–Rosenberg [1]. 2010 Mathematics Subject Classification: Primary: 53C42; Secondary: 53B25.


Introduction and statement of the main result
In recent years, following the seminal ideas established by Reilly in [24,25], several authors obtained geometric properties of a hypersurface through the study of the Euler-Lagrange equation associated to certain variational problems (see, for instance, [1,4,5,6,14,11,29]).Proceeding into this branch, in this paper we deal with a suitable class of closed hypersurfaces immersed in the Euclidean space R n+1 , which are critical points for the variational problem of minimizing a linear combination of certain area functions preserving the volume enclosed.A precise description of our object of study will be given after some preliminaries.
Let F : S n → R + be a positive smooth function which satisfies the following convexity condition: where D 2 F denotes the intrinsic Hessian of F on the n-sphere S n of R n+1 , I denotes the identity on T x S n , and > 0 means that the matrix is positive definite.We consider the map whose image W F = φ(S n ) is a smooth, convex hypersurfaces in R n+1 called the Wulff shape of F (for more details concerning the properties of the Wulff shape see, for instance, [9,14,16,17,18,19,27]).We note that, when F ≡ 1 we have that the Wulff shape of F is just the n-dimensional Euclidean sphere S n ⊂ R n+1 .Throughout this paper, x : M n → R n+1 will stand for a smooth immersion of a closed oriented hypersurface and N : M n → S n will denote its corresponding Gauss map.In this setting, let According to the previous definitions, we note that although A F and dN be symmetric operators, S F is symmetric if and only if A F and dN commute, which does not occur in general.We point out that all the roots of the characteristic polynomial of S F are real.The eigenvalues of S F are called the anisotropic principal curvatures of x and are denoted by λ 1 , . . ., λ n .Moreover, if the principal curvatures of x are positive, so are the anisotropic principal curvatures λ i (for a proof, see Lemma 2 in [10]).
At each p ∈ M n , S F restricts to a linear map S F (p) : where S 0 ≡ 1 by construction.If p ∈ M n and {λ k } are the eigenvalues with respect to the operator S F (p), one immediately sees that There exist many works concerning the properties of the k-th anisotropic mean curvatures of hypersurfaces in R n+1 space.We refer the readers to [10,13,14,15,16,23,30].
At this point, we are in a position to define our geometrical object of study: an immersion x : M n → R n+1 is said a (r, s, F )-linear Weingarten hypersurface if, for some integers r and s satisfying the inequality 0 ≤ r ≤ s ≤ n − 1, holds the following linear relation , for some nonnegative real numbers a k , k ∈ {r, . . ., s}, with at least one non zero, where b k = (k + 1) n k+1 and F : S n → R + is a positive smooth function which satisfies the convexity condition (1.1).
We observe that, when r = 0, s = 1, and F = 1, these hypersurfaces are classically called linear Weingarten hypersurfaces and, in the last years, a vast literature has been produced in the direction to obtain characterization results of them (see, for instance, [2,3,7,8,20,26]).In this paper we extend this study to the anisotropic case.It is said that a phenomenon has anisotropic behavior when its effects vary with the direction; as opposed to isotropic behavior (homogeneous in all directions).This is observed in many phenomena of nature.Indeed, it appears that anisotropic properties are present in the study of many phenomena.For example, crystals whose propagation of light depends on the direction exhibit optical anisotropy [28].The (s, s, F )-linear Weingarten hypersurfaces are exactly the hypersurfaces with H F s+1 constant.On the other hand, taking into account that all the anisotropic principal curvatures of the Wulff shape W F are constant (see, for example, [14]), we have that W F constitutes a natural example of closed (r, s, F )-linear Weingarten hypersurface immersed in R n+1 , for any 0 ≤ r ≤ s ≤ n − 1.
Motivated by the previous discussion, here we will establish the notion of stability concerning closed (r, s, F )-linear Weingarten hypersurfaces in R n+1 .Such concept arises considering the variational problem of minimizing a suitable linear combination of certain functionals (k, F )-th areas for volume-preserving variations (cf.Section 4).Now, we are in position to state our main result.
Theorem 1.Let r and s be integers satisfying 0 ≤ r ≤ s ≤ n − 2, n ≥ 3, and let x : M n → R n+1 be a closed (r, s, F )-linear Weingarten hypersurface with H F s+1 positive.Then, x : M n → R n+1 is stable if, and only if, up to translations and homotheties, x(M ) is the Wulff shape of F .
We observed that taking r = s in Theorem 1 we reobtain Theorem 1.3 of [14].The proof of Theorem 1 is given in Section 5.
Remark 1. Related to the isotropic case, Micallef and Moore [22] proved that any minimal 2-sphere in a manifold with positive isotropic curvature is unstable.More recently, Li [21] showed the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to C in any complete orientable 4-dimensional Riemannian manifold with uniformly positive isotropic curvature.

The operators P k , T k , and L k
In order to give a description of our variational problem, we will need to define some suitable operators associated to a hypersurface x : M n → R n+1 as in the previous section.The first ones are the operators P k : X(M ) → X(M ), 0 ≤ k ≤ n, which can be defined inductively from the F -Weingarten operator S F and k-th anisotropic mean curvatures H F k by (2.1) where I denotes the identity in X(M ).Equivalently, (2.2) Note that each P k (p) is also a linear operator on each tangent space T p M which commutes with S F (p).The operator Note that, since A F and dN are symmetric, from (2.2) we have that the transformations T k are all self-adjoint operators.So, likewise S F • AF , dN • S F , and dN • P k are symmetric.Moreover, taking into account that Finally, we define the operator Equivalently, where we denote the coefficients of covariant differential of f and T k with respect to a (local) orthonormal frame {e 1 , . . ., e n } on M n by f i and (T k ) ij , respectively.In particular, we the F -Laplacian is defined by

Key lemmas
This section is devoted to other auxiliary results which will be also necessary to prove Theorem 1.Initially, we recall suitable inequalities concerning the k-th anisotropic mean curvatures, which will be very useful to show our main result.Lemma 1.Let x : M n → R n+1 be a closed hypersurface and F : S n → R + satisfying the condition (1.1).If H F s+1 is positive on M n then for 1 ≤ k ≤ s, we have: Moreover equality holds for some k in (ii) if, and only if, the anisotropic principal curvatures are equal.
Proof: Item (i) corresponds to Lemma 10 of [10].For (ii), it is known that the following generalization of the Cauchy-Schwarz type inequality holds true (see, for instance, [12, Theorem 51, p. 52, and Theorem 144, p. 104]) for any 1 ≤ k ≤ n − 1: ≥ 0, the equality occurring for some k if, and only if, at this point the anisotropic principal curvatures are equal.As H F k > 0 for any 0 ≤ k ≤ s + 1, we can write the inequality (3.1) as follows: for any 1 ≤ k ≤ s + 1, and equality holds for some k if, and only if, at this point the anisotropic principal curvatures are equal.Hence, we have the inequalities for all 0 ≤ k ≤ s, and equality holds in (3.2) if, and only if, the anisotropic principal curvatures are equal.Thus, from (3.2) we obtain the results, finishing the proof.Now, we will quote well known result which will be used later (for a proof, see He and Li [13,15] or Palmer [23]).Lemma 2. Let x : M n → R n+1 be an isometric immersion of a closed orientable Riemannian manifold M n and F : then up to translations and homotheties, x(M ) is the Wulff shape of F .
In the next lemma, we recall the so-called Minkowski formulas (see He and Li [13,15]).Lemma 3. Let x : M n → R n+1 be a closed hypersurface and F : S n → R + satisfying the condition (1.1).For each 0 ≤ j ≤ n − 1, the following Minkowski-type formulas hold At this point, we will fix some notation.Given f ∈ C ∞ (M ) smooth function, we define: We will also need of the following result due to He and Li [13].
Lemma 4. For each 0 ≤ k ≤ n − 1, we have: (3.5) Finally, we also need of the following symmetry result, which will be crucial to establish the proof of Theorem 1: Proof: By (3.4) we have By Stokes theorem and from the symmetry of T k , we have (3.8) Thus, by (3.3) and (3.8) (3.9) Therefore, by (3.7), (3.9), and (3.4), we have finishing the proof.

Description of the variational problem
A variation of a closed oriented hypersurface x : M n → R n+1 is a smooth map X : M n × (− , ) −→ R n+1 such that, for all t ∈ (− , ), the map X t : M n → R n+1 given by X t (p) = X(t, p) is an immersion such that X 0 = x.In what follows, dM t will denote the volume element of the metric induced on M n by X t and N t will stand for the unit normal vector field along X t .
The variational field associated to the variation X is the vector field ∂X ∂t t=0 .In this setting, denoting by (•) the tangential component on M n , we have that The volume of the variation X is the functional and we say X is volume-preserving if V(t) = V(0), for all t ∈ (− , ).The next lemma is a well known result and a proof of it can be found in [6].
In particular, X is volume-preserving if, and only if, M f dM t = 0 for all t ∈ (− , ).
We can reason as in the proof of Lemma 7 of [10] to get the following: there exists a volume-preserving variation X : M n × (− , ) → R n+1 of x such that its variational vector field is f N .
For each k ∈ {0, . . ., n}, we define the (k, F )-area functional A k,F : (− , ) → R associated to the variation X : Note that for F ≡ 1 and k = 0, it is the classical area functional.
The result below follows from Lemma 3.1 of [14].
where f is defined in (4.1).
The previous lemma allows us to obtain the first variation of the (k, F )-area functional (cf.Theorem 3.3 of [14]).
where f is defined in (4.1).Now, motivated by the concept of (r, s, F )-linear Weingarten hypersurface, it is natural to consider the variational problem of minimizing the following functional for all variations X : The Jacobi functional associated to this variational problem is given by where λ is a constant to be determined.As an immediate consequence of Lemmas 6 and 9 we get In order to make an appropriated choice of λ, let We point out to the fact that, in case and this notation will be used in what follows without further comments.Hence, choosing λ = H, we arrive at From (4.9) we observe that the critical points of the variational problem described above are exactly the closed (r, s, F )-linear Weingarten hypersurfaces.This fact allow us to define a closed (r, s, F )-linear Weingarten hypersurface x : M n → R n+1 being stable when B r,s,F (0) ≥ 0, for all volume-preserving variations X : Furthermore, we can reason as in [6] to obtain the following stability criterion: a closed (r, s, F )-linear Weingarten hypersurface x : M n → R n+1 is stable if, and only if, J r,s,F (0 The sought formula for the second variation of J r,s,F is a straightforward consequence of Lemmas 8 and 9. Proof: From (4.9) we obtain Consequently, taking into account (4.4) and (4.8), we get To finish the proof, we observe that the above expression depends only on the hypersurface x : M n → R n+1 and on the function f ∈ C ∞ (M ).

Proof of Theorem 1
In what follows, we will consider the set If x(M ) is (up to translations and homotheties) the Wulff shape of F , from the proof of Theorem 1.3 in [14] we have that So, from (4.10), J r,s,F (f ) ≥ 0 for all f ∈ G. Therefore, x is stable.
Reciprocally, supposing that x is stable, we have J r,s,F (f ) ≥ 0, for all f ∈ G.Moreover, from Lemma 3 we can choose f = γF (N ) + ξ x, N as the test function, where We also note that, from (3.3), (3.4), and Proposition 1 we have that On the other hand, from (3.5) and (3.6) we get Hence, we get Furthermore, from Lemma 5 we also have that (5.2) But, from (5.1) and Lemma 4 we have Therefore, from (5.2), (5.3), (5.4) we have Thus, from Lemma 3 we get Now, for each point p ∈ M n , we consider the polynomial in z given by P k,p (z) = H F 1 H F k+1 z 2 − 2ξH F k+1 z + ξ 2 H F k .In this case, we have that the discriminant of P k,p is where the last inequality in the expression above is given by Lemma 1.Thus, we have that P k,p ≥ 0, ∀ p ∈ M n , and ∀ k ∈ {r, . . ., s}.In particular, (5.6) γ) ≥ 0, ∀ p ∈ M n and ∀ k ∈ {r, . . ., s}.So, from (5.5), (5.6), and Lemma 1, we have (5.7) From (5.7), since x is stable and taking into account that each therm of J r,s (0)(f ) is nonpositive, we easily see that they are, in fact, identically zero.Consequently, using once more (5.7),we get that H F 1 H F k+1 − H F k+2 = 0 on M n , for all r ≤ k ≤ s.So, from item (ii) of Lemma 1, we have that the anisotropic principal curvatures are equal.Therefore, Lemma 2 assures us that, up to translations and homotheties, x(M ) is the Wulff shape of F , finishing the proof of Theorem 1.
Remark 2. Concerning our constraint on the sign of H F s+1 in Theorem 1, we observe that it is crucial to the success of our technique, since the Gardin type inequalities of Lemma 1 constitutes our algebraic tool to detect the anisotropic umbilicity of the closed (r, s, F )-linear Weingarten hypersurface.We also note that this phenomenon already occurs in the context of the works of Alencar-do Carmo-Rosenberg [1] and Barbosa-Colares [6].

(1. 3 )
A F := D 2 F + F I and N F := φ • N : M n −→ W F , which is called the generalized Gauss map into the Wulff shape.Then S F := −dN F = −A F • dN is defined as being the F -Weingarten operator.