STABILITY OF HYPERSURFACES WITH CONSTANT (r +1)-TH ANISOTROPIC MEAN CURVATURE AND HAIZHONG LI

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1 Illinois Journal of athematics Volume 52, Number 4, Winter 2008, Pages S STABILITY OF HYPERSURFACES WITH CONSTANT (r +1)-TH ANISOTROPIC EAN CURVATURE YIJUN HE * AND HAIZHONG LI Abstract. Given a positive function F on S n which satisfies a convexity condition, we define the r-th anisotropic mean curvature function Hr F for hypersurfaces in R n+1 which is a generalization of the usual r-th mean curvature function. Let X : R n+1 be an n-dimensional closed hypersurface with Hr+1 F = constant, for some r with 0 r n 1, which is a critical point for a variational problem. We show that X() is stable if and only if X() isthewulffshape. 1. Introduction Let F : S n R + be a smooth function which satisfies the following convexity condition: (1.1) (D 2 F + F 1) x > 0 x S n, where S n denotes the standard unit sphere in R n+1, D 2 F denotes the intrinsic Hessian of F on S n and 1 denotes the identity on T x S n, > 0 means that the matrix is positive definite. We consider the map (1.2) φ : S n R n+1, x F (x)x + (grad S n F ) x, its image W F = φ(s n ) is a smooth, convex hypersurface in R n+1 called the Wulff shape of F (see [4], [7] [9], [11], [14], [18], [19]). We note when F 1, W F is just the sphere S n. Received December 15, 2007; received in final form January 23, * Corresponding author. The first author was partially supported by Youth Science Foundation of Shanxi Province, China (Grant No ). The second author was partially supported by the grant No of the NSFC and by SRFDP athematics Subject Classification. Primary 53C42, 53A10. Secondary 49Q c 2009 University of Illinois

2 1302 Y. HE AND H. LI Now let X : R n+1 be a smooth immersion of a closed, orientable hypersurface. Let ν : S n denotes its Gauss map, that is ν is the unit inner normal vector of. Let A F = D 2 F + F 1, S F = d(φ ν) = A F dν. S F is called the F - Weingarten operator, and the eigenvalues of S F are called anisotropic principal curvatures. Let σ r be the elementary symmetric functions of the anisotropic principal curvatures λ 1,λ 2,...,λ n : (1.3) σ r = λ i1 λ ir (1 r n). i 1< <i r We set σ 0 =1. The r-th anisotropic mean curvature Hr F is defined by Hr F = σ r /Cn, r also see Reilly [16]. For each r, 0 r n 1, we set (1.4) A r,f = F (ν)σ r da X. The algebraic (n + 1)-volume enclosed by is given by (1.5) V = 1 X,ν da X. n +1 We consider those hypersurfaces which are critical points of A r,f restricted to those hypersurfaces enclosing a fixed volume V. By a standard argument involving Lagrange multipliers, this means we are considering critical points of the functional (1.6) F r,f ;Λ = A r,f +ΛV (X), where Λ is a constant. We will show the Euler Lagrange equation of F r,f ;Λ is: (1.7) (r +1)σ r+1 Λ=0. So the critical points are just hypersurfaces with Hr+1 F = constant. If F 1, then the function A r,f is just the functional A r = S r da X which was studied by Alencar, do Carmo, and Rosenberg in [1], where H r = S r /Cn r is the usual r-th mean curvature. For such a variational problem, they call a critical immersion X of the functional A r (that is, a hypersurface with H r+1 = constant) stable if and only if the second variation of A r is nonnegative for all variations of X preserving the enclosed (n +1)-volume V.Theyproved the following theorem. Theorem 1.1 ([1]). Suppose 0 r n 1. Let X : R n+1 be a closed hypersurface with H r+1 = constant. Then X is stable if and only if X() is a round sphere. Analogously, we call a critical immersion X of the functional A r,f stable if and only if the second variation of A r,f (or equivalently of F r,f ;Λ )is nonnegative for all variations of X preserving the enclosed (n +1)-volume V.

3 STABILITY OF HYPERSURFACES 1303 In [14], Palmer proved the following theorem (also see Winklmann [19]). Theorem 1.2 ([14]). Let X : R n+1 be a closed hypersurface with H1 F = constant. Then X is stable if and only if, up to translations and homotheties, X() is the Wulff shape. In this paper, we prove the following theorem. Theorem 1.3. Suppose 0 r n 1. Let X : R n+1 be a closed hypersurface with Hr+1 F = constant. Then X is stable if and only if, up to translations and homotheties, X() is the Wulff shape. Remark 1.4. In the case F 1, Theorem 1.3 becomes Theorem 1.1. Theorem 1.3 gives an affirmative answer to the problem proposed in [8]. We also note that in the case F 1, our result here gives a new and geometric proof of Theorem 1.1, which is different from [1]. 2. Preliminaries Let X : R n+1 be a smooth closed, oriented hypersurface with Gauss map ν : S n,thatis,ν is the unit inner normal vector field. Let X t be a variation of X, andν t : S n be the Gauss map of X t. We define (2.1) ψ = dxt dt,ν t, ξ = ( dxt dt where represents the tangent component and ψ, ξ are dependent of t. The corresponding first variation of the unit normal vector is given by (see [8], [11], [14], [19]) (2.2) ν t = grad ψ +dν t (ξ), the first variation of the volume element is (see [2], [3], or [10]) (2.3) t da Xt =(divξ nhψ)da Xt, and the first variation of the volume V is (2.4) V (t)= ψ da Xt, where grad, div, H represents the gradients, the divergence, the mean curvature with respect to X t, respectively. Let {E 1,...,E n } be a local orthogonal frame on S n,lete i = e i (t)=e i ν t, where i =1,...,n and ν t is the Gauss map of X t,then{e 1,...,e n } is a local orthogonal frame of X t : R n+1. The structure equations of x : S n R n+1 are: dx = i θ ie i, de i = j (2.5) θ ije j θ i x, dθ i = j θ ij θ j, dθ ij k θ ik θ kj = 1 2 ), k,l R ijkl θ k θ l = θ i θ j,

4 1304 Y. HE AND H. LI where θ ij + θ ji =0 and R ijkl = δ il δ jk δ ik δ jl. The structure equations of X t are (see [12], [13]): dx t = i ω ie i, dν t = h ijω j e i, (2.6) de i = j ω ije j + j h ijω j ν t, dω i = j ω ij ω j, dω ij k ω ik ω kj = 1 2 k,l R ijklθ k θ l, where ω ij + ω ji =0, R ijkl + R ijlk =0, and R ijkl are the components of the Riemannian curvature tensor of X t () with respect to the induced metric dx t dx t. Here, we have omitted the variable t for some geometric quantities. From de i =d(e i ν t )=νt de i = j ν t θ ij e j νt θ i ν t, we get { ω ij = νt θ ij, (2.7) νt θ i = j h ijω j, where ω ij + ω ji =0, h ij = h ji. Let F : S n R + be a smooth function, we denote the coefficients of covariant differential of F,grad S n F with respect to {E i } i=1,...,n by F i,f ij respectively. From (2.7), d(f (ν t )) = ν t df = ν t ( i F iθ i )= (F i ν t )h ij ω j,thus, (2.8) grad(f (ν t )) = (F i ν t )h ij e j =dν t (grad S n F ). Through a direct calculation, we easily get (2.9) dφ =(D 2 F + F 1) dx = A ij θ i E j, where A ij is the coefficient of A F,thatis,A ij = F ij + Fδ ij. Taking exterior differential of (2.9) and using (2.5), we get (2.10) A ijk = A jik = A ikj, where A ijk denotes coefficient of the covariant differential of A F on S n. We define (A ij ν t ) k by (2.11) d(a ij ν t )+ (A kj ν t )ω ki + k (A ik ν t )ω kj = k (A ij ν t ) k ω k. By a direct calculation by using (2.7) and(2.11), we have (2.12) (A ij ν t ) k = l h kl A ijl ν t.

5 STABILITY OF HYPERSURFACES 1305 We define L ij by (2.13) ( ) dei = L ij e j, dt j where denote the tangent component, then L ij = L ji andwehave(see [2], [3], or [10]) (2.14) h ij = ψ ij + k {h ijk ξ k + ψh ik h jk + h ik L kj + h jk L ki }. Let s ij = k (A ik ν)h kj,thenfrom(2.7) and(2.9), we have (2.15) d(φ ν t )=ν t dφ = s ij ω j e i. We define S F by S F = d(φ ν)= A F dν, thenwehaves F (e j )= i s ije i. We call S F the F -Weingarten operator. From the positive definiteness of (A ij ) and the symmetry of (h ij ), we know the eigenvalues of (s ij )areall real. We call them anisotropic principal curvatures, and denote them by λ 1,...,λ n. Taking exterior differential of (2.15) and using (2.6), we get (2.16) s ijk = s ikj, where s ijk denotes coefficient of the covariant differential of S F. We have n invariants, the elementary symmetric function σ r of the anisotropic principal curvatures: (2.17) σ r = λ i1 λ in i 1< <i r (1 r n). For convenience, we set σ 0 =1 and σ n+1 =0. The r-th anisotropic mean curvature Hr F is defined by (2.18) Hr F = σ r /Cn, r Cn r n! = r!(n r)!. We have, by use of (2.2) and(2.6), d((a ij E i E j ) ν t ) = ( (2.19) D(Aij E i E j ) ) dt ν t,ν t = ( A ijk ψ k + h kl ξ l )e i e j,,k l where D is the Levi Civita connection on S n.

6 1306 Y. HE AND H. LI On the other hand, we have d((a ij E i E j ) ν t ) (2.20) dt = { ( ) ( ) } A dei dej ije i e j + A ij e j + A ije i. dt dt By use of (2.13), we get from (2.19) and(2.20) (2.21) d(a ij ν t ) dt = A ij(t) = { A ijk ψ k A ijk h kl ξ l + A ik L kj + A jk L ki }. k l By (2.12), (2.14), (2.21) and the fact L ij = L ji, through a direct calculation, we get the following lemma. Lemma 2.1. ds ij = s dt ij(t)= k {(A ik ψ k ) j + s ijk ξ k + ψs ik h kj + s kj L ki + s ik L kj }. As is a closed oriented hypersurface, one can find a point where all the principal curvatures with respect to ν are positive. By the positiveness of A F, all the anisotropic principal curvatures are positive at this point. Using the results of Gårding [5], we have the following lemma. Lemma 2.2. Let X : R n+1 be a closed, oriented hypersurface. Assume Hr+1 F > 0 holds at every point of, then Hk F > 0 holds on every point of for every k =1,...,r. Using the characteristic polynomial of S F, σ r is defined by n (2.22) det(ti S F )= ( 1) r σ r t n r. So, we have (2.23) σ r = 1 r! r=0 i 1,...,i r;j 1,...,j r δ j1 jr i 1 i r s i1j 1 s irj r, where δ j1 jr i 1 i r is the usual generalized Kronecker symbol, i.e., δ j1 jr i 1 i r equals +1 (resp. 1) if i 1 i r are distinct and (j 1 j r )isaneven(resp.odd) permutation of (i 1 i r ) and in other cases it equals zero. We introduce two important operators P r and T r by (2.24) P r = σ r I σ r 1 S F + +( 1) r SF r, r=0, 1,...,n, (2.25) T r = P r A F, r=0, 1,...,n 1.

7 STABILITY OF HYPERSURFACES 1307 Obviously, P n =0 andwehave (2.26) P r = σ r I P r 1 S F = σ r I + T r 1 dν, r =1,...,n. From the symmetry of A F and dν, S F A F and dν S F are symmetric, so T r = P r A F and dν P r are also symmetric for each r. Lemma 2.3. The matrix of P r is given by: (2.27) (P r ) ij = 1 r! i 1,...,i r;j 1,...,j r δ j1 jri i 1 i rj s i 1j 1 s irj r. Proof. We prove Lemma 2.3 inductively. For r = 0, it is easy to check that (2.27) istrue. Assume (2.27) istrueforr = k, thenfrom(2.26), (P k+1 ) ij = σ k+1 δj i (P k ) il s lj l 1 ( = δ j1 j k+1 i (k + 1)! 1 i k+1 δj i l δ j1 j l 1ij l+1 j k+1 i 1 i l 1 i l i l+1 i k+1 δ j l j ) = s i1j 1 s ik+1 j k+1 1 j δ 1 j k+1 i i (k + 1)! 1 i k+1 j s i 1j 1 s ik+1 j k+1. Lemma 2.4. For each r, we have (i) j (P r) jij =0, (ii) tr(p r S F )=(r +1)σ r+1, (iii) tr(p r )=(n r)σ r, (iv) tr(p r S 2 F )=σ 1σ r+1 (r +2)σ r+2. Proof. We only prove (i), the others are easily obtained from (2.23), (2.26), and (2.27). Noting s i1j 1 s irj rj = s i1j 1 s irjj r by (2.16) andδ j1 jrj i = δj1 jjr 1 i ri i,we 1 i ri have 1 (P r ) jij = δ j1 jrj i (r 1)! s 1 i ri i 1j 1 s irj rj =0. j i 1,...,i r;j 1,...,j r;j Remark 2.5. When F = 1, Lemma 2.4 was a well-known result (for example, see Barbosa Colares [2], Reilly [15], or Rosenberg [17]). Since P r 1 S F is symmetric and L ij is anti-symmetric, we have (2.28) (P r 1 ) ji (s kj L ki + s ik L kj )=0.,k

8 = 1308 Y. HE AND H. LI From (2.16), (2.26), and (i) of Lemma 2.4, we get (2.29) (σ r ) k = j (σ r δ jk ) j = j (P r ) jkj + j,l [(P r 1 ) jl s lk ] j = (P r 1 ) ji s ijk. 3. First and second variation formulas of F r,f ;Λ Define the operator L r : C () C () as follows: (3.1) L r f = [(T r ) ij f j ] i. Lemma 3.1. dσ r dt = σ r(t)=l r 1 ψ + ψ T r 1 dν t, dν t + grad σ r,ξ. Proof. Using (2.23), (2.28), (2.29), Lemma 2.1, Lemma 2.3, and (i) of Lemma 2.4, wehave σ r 1 = δ j1 jr i (r 1)! 1 i r s i1j 1 s ir 1j r 1 s i rj r i 1,...,i r;j 1,...,j r = (P r 1 ) ji s ij =,k(p r 1 ) ji [(A ik ψ k ) j + ψs ik h kj + s ijk ξ k + s kj L ki + s ik L kj ] = [(P r 1 ) ji A ik ψ k ] j + ψ r 1 ) ji A il h lk h kj +,k,k,l(p k (σ r ) k ξ k = j,k [(T r 1 ) jk ψ k ] j + ψ,k (T r 1 ) ji h ik h kj + k (σ r ) k ξ k = L r 1 ψ + ψ T r 1 dν t, dν t + grad σ r,ξ. Lemma 3.2. For each 0 r n, we have (3.2) div ( ) P r (grad S n F ) ν t + F (νt )tr(p r dν t )= (r +1)σ r+1 and (3.3) div(p r X )+ X,ν t tr(p r dν t )=(n r)σ r. Proof. From (2.6), (2.15), and Lemma 2.4, div ( ) ( P r (grad S n F ) ν t =div Pr (φ ν t ) ) ( (Pr ) ji φ ν t,e i ) j

9 STABILITY OF HYPERSURFACES 1309 = (P r ) ji s ij + F (ν t ) (P r ) ji h ij = tr(p r S F ) F (ν t )tr(p r dν t ) = (r +1)σ r+1 F (ν t )tr(p r dν t ), div(p r X )= ((P r ) ji X,e i ) j = (P r ) ji δ ij + (P r ) ji h ij X,ν t =tr(p r ) tr(p r dν t ) X,ν t =(n r)σ r tr(p r dν t ) X,ν t. Thus, the conclusion follows. Theorem 3.3 (First variational formula of A r,f ). (3.4) A r,f (t)= (r +1) ψσ r+1 da Xt. Proof. We have (F (ν t )) = grad S n F,ν t, so by use of Lemma 3.1, Lemma 3.2, (2.2), (2.3), (2.8), (2.26), and Stokes formula, we have A r,f ( (t)= F (νt )σ r +(F (ν t )) ) σ r daxt + F (ν t )σ r t da Xt = {F (ν t )div(t r 1 grad ψ)+f (ν t )ψ T r 1 dν t, dν t + F (ν t ) grad σ r,ξ + σ r (grad S n F ) ν t, grad ψ +dν t (ξ) + F (ν t )σ r ( nhψ +divξ)} da Xt { = grad(f (νt )),T r 1 grad ψ + F (ν t )ψ T r 1 dν t, dν t + F (ν t )gradσ r,ξ + ψ div ( ) σ r (grad S n F ) ν t + σ r grad(f (ν t )),ξ nhψf (ν t )σ r + F (ν t )σ r div ξ } da Xt { = Tr 1 grad(f (ν t )), grad ψ + F (ν t )ψ T r 1 dν t, dν t + ψ div ( ) } σ r (grad S n F ) ν t nhψf (νt )σ r daxt = ψ { div ( ) σ r (grad S n F ) ν t +div(tr 1 grad(f (ν t ))) } + F (ν t ) T r 1 dν t, dν t nhf (ν t )σ r daxt = ψ{div[(σ r + T r 1 dν t )(grad S n F ) ν t ] + F (ν t )tr[(t r 1 dν t + σ r I) dν t ]} da Xt

10 1310 Y. HE AND H. LI = ψ { div ( ) P r (grad S n F ) ν t + F (νt )tr(p r dν t ) } da Xt = (r +1) ψσ r+1 da Xt. Remark 3.4. When F = 1, Lemma 4.1 and Theorem 3.3 were proved by R. Reilly [15] (also see [2], [3]). From (1.6), (2.4), and (3.4), we get Proposition 3.5 (The first variational formula). For all variations of X, we have (3.5) F r,f ;Λ(t)= ψ{(r +1)σ r+1 Λ} da Xt. Hence, we obtain the Euler Lagrange equation of F r,f ;Λ : (3.6) (r +1)σ r+1 Λ=0. Theorem 3.6 (The second variational formula). Let X : R n+1 be an n-dimensional closed hypersurface, which satisfies (3.6), then for all variations of X preserving V, the second variational formula of A r,f at t =0is given by (3.7) A r (0) = F r,f ;Λ(0) = (r +1) ψ{l r ψ + ψ T r dν, dν } da X, where ψ satisfies (3.8) ψ da X =0. Proof. Differentiating (3.5), we get (3.7)byuseof(3.6) and Lemma 3.1. We call X : R n+1 to be a stable critical point of A r,f for all variations of X preserving V, if it satisfies (3.6) anda r (0) 0 for all ψ with condition (3.8). 4. ProofofTheorem1.3 Firstly, we prove that if X() is, up to translations and homotheties, the Wulff shape, then X is stable. From dφ =(D 2 F + F 1) dx, dφ is perpendicular to x. So ν = x is the unit inner normal vector. We have (4.1) dφ = A F dν = A jk h ki ω i e j. ijk On the other hand, (4.2) dφ = i ω i e i,

11 STABILITY OF HYPERSURFACES 1311 so we have (4.3) s ij = k A ik h kj = δ ij. From this, we easily get σ r = Cn r and σ r+1 = Cn r+1, thus, the Wulff shape satisfies (3.6) withλ=(r +1)Cn r+1. Through a direct calculation, we easily know for Wulff shape, (4.4) A r (0) = (r +1)Cn 1 r [div(a F grad ψ)+ψ A F dν, dν ]da X, and ψ satisfies (4.5) ψ da X =0. From Palmer [14] (also see Winklmann [19]), we know A r (0) 0, that is the Wulff shape is stable. Next, we prove that if X is stable, then up to translations and homotheties, X() is the Wulff shape. We recall the following lemmas. Lemma 4.1 ([7], [8]). For each r =0, 1,...,n 1, the following integral formulas of inkowski type hold: ( (4.6) H F r F (ν)+hr+1 X,ν F ) da X =0, r=0, 1,...,n 1. Lemma 4.2 ([7], [8], [14]). If λ 1 = λ 2 = = λ n = const 0, then up to translations and homotheties, X() is the Wulff shape. From Lemmas 4.1 and (3.8), we can choose ψ = αf (ν)+h F r+1 X,ν as the test function, where α = F (ν)hf r da X / F (ν)da X. For every smooth function f : R, and each r, we define: (4.7) I r [f]=l r f + f T r dν, dν. Then we have from (3.7) (4.8) A r (0) = (r +1) Lemma 4.3. For each 0 r n 1, we have ψi r [ψ]da X. (4.9) I r [F ν]= grad σ r+1, (grad S n F ) ν + σ 1 σ r+1 (r +2)σ r+2 and (4.10) I r [ X,ν ]= grad σ r+1,x (r +1)σ r+1. Proof. From (2.8) and(2.26), I r [F ν]=div{t r grad(f (ν))} + F (ν) T r dν, dν =div ( T r dν(grad S n F ) ν ) + F (ν) T r dν, dν =div ( P r+1 (grad S n F ) ν ) + F (ν)tr(p r+1 dν)

12 1312 Y. HE AND H. LI grad σ r+1, (grad S n F ) ν { ( σ r+1 div P0 (grad S n F ) ν ) + F (ν)tr(p 0 dν) }, I r [ X,ν ]=div(t r grad X,ν )+ X,ν T r dν, dν =div(t r dνx )+ X,ν T r dν, dν =div(p r+1 X )+ X,ν tr(p r+1 dν) grad σ r+1,x σ r+1 {div(p 0 X )+ X,ν tr(p 0 dν)}. So the conclusions follow from Lemma 3.2. As H F r+1 is a constant, from (4.9) and(4.10), we have (4.11) I r [ψ]=αi r [F ν]+h F r+1i r [ X,ν ] = α ( σ 1 σ r+1 (r +2)σ r+2 ) (r +1)H F r+1 σ r+1 = C r+1 n {α[nh F 1 H F r+1 (n r 1)H F r+2] (r + 1)(H F r+1) 2 }. Therefore, we obtain from Lemma 4.1 (recall H F r+1 is constant and ψ da X = 0) 1 r +1 A r (0) = ψi r [ψ]da X = ψcn r+1 {α[nh1 F Hr+1 F (n r 1)Hr+2] F (r + 1)(Hr+1) F 2 } da X = αcn r+1 [αf (ν)+hr+1 X,ν ][nh F 1 F Hr+1 F (n r 1)Hr+2]dA F X = α 2 Cn r+1 F (ν)[nh1 F Hr+1 F (n r 1)Hr+2]dA F X αcn r+1 Hr+1 F X,ν [nh1 F Hr+1 F (n r 1)Hr+2]dA F X = α 2 Cn r+1 F (ν)[nh1 F Hr+1 F (n r 1)Hr+2]dA F X + αcn r+1 Hr+1 F F (ν)[nhr+1 F (n r 1)Hr+1]dA F X = α 2 (n r 1)Cn r+1 F (ν)(h1 F Hr+1 F Hr+2)dA F X α(r +1)Cr+1 n (H F r+1) 2 F (ν)da X { ( F (ν)h1 F da X F (ν) HF r Hr+1 F da X F (ν)da X ) 2 },

13 STABILITY OF HYPERSURFACES 1313 whereweusedα = F (ν)hf r da X / F (ν)da X in the last equality of the above formula. As Hr+1 F is a constant, it must be positive by the compactness of. Thus, by Lemma 2.2, H1 F,...,Hr F are all positive. So, from [6] or[20], we have: (i) for each 0 r<n 1, (4.12) H F 1 H F r+1 H F r+2 0, with the equality holds if and only if λ 1 = = λ n,and (ii) for each 1 r n 1, ( ) 2 F (ν)h1 F da X F (ν) HF r (4.13) Hr+1 F da X F (ν)da X ( ) 2 F (ν)h1 F da X F (ν)/h1 F da X F (ν)da X 0, with the equality holds if and only if λ 1 = = λ n. From (4.12) and(4.13), we easily obtain that, for each 0 r n 1, A r (0) 0, with the equality holds if and only if λ 1 = = λ n. Thus, from Lemma 4.2, up to translations and homotheties, X() is the Wulff shape. We complete the proof of Theorem 1.3. Acknowledgment. The authors would like to express their thanks to the referee for helpful comments. References [1] H. Alencar,. do Carmo and H. Rosenberg, On the first eigenvalue of the linearized operator of the r-th mean curvature of a hypersurface, Ann. Global Anal. Geom. 11 (1993), R [2] J.L..BarbosaandA.G.Colares,Stability of hypersurfaces with constant r-mean curvature, Ann. Global Anal. Geom. 15 (1997), R [3] L. F. Cao and H. Li, r-minimal submanifolds in space forms, Ann. Global Anal. Geom. 32 (2007), R [4] U. Clarenz, The Wulff-shape minimizes an anisotropicwillmore functional, Interfaces Free Bound. 6 (2004), R [5] L. Gårding, An inequality for hyperbolic polynomials, J. ath. ech. 8 (1959), R [6] G.H.Hardy,J.E.LittlewoodandG.Polya,Inequalities, Reprint of the 1952 edition, Cambridge athematical Library, Cambridge University Press, Cambridge, [7] Y. J. He and H. Li, Integral formula of inkowski type and new characterization of the Wulff shape, Acta ath. Sin. (Engl. Ser.) 24 (2008), R [8] Y. J. He and H. Li, A new variational characterization of the Wulff shape, Differential Geom. App. 26 (2008), R [9] Y. J. He, H. Li, H. a and J. Q. Ge, Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures, Indiana Univ. ath. J. 58 (2009),

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