Alexandrov-Fenchel type inequalities in the sphere
aa r X i v : . [ m a t h . DG ] J a n ALEXANDROV-FENCHEL TYPE INEQUALITIES IN THESPHERE
MIN CHEN AND JUN SUN
Abstract.
In this paper, we attempt to use two types of flows to study therelations between quermassintegrals A k (see Definition 1.1), which correspondto the Alexandrov-Fenchel inequalities for closed convex C -hypersurfaces in S n +1+ . Introduction
The Alexandrov-Fenchel inequalities [1, 2] for the quermassintegrals of convexdomains in R n +1 are fundamental in classical geometry. In [11], Guan and Li ex-tended these inequalities to star-shaped domains in R n +1 . There have been exten-sive interests on studying the Alexandrov-Fenchel type inequalities for quermass-integral in space forms. Let N n +1 ( K ) be the space form with constant sectionalcurvature K = 1, 0 or −
1. Under the Gaussian geodesic normal coordinates, themetric can be expressed as ds = dρ + φ ( ρ ) dz , where φ ( ρ ) = sin ρ, ρ ∈ [0 , π ) when K = 1; φ ( ρ ) = ρ, ρ ∈ [0 , ∞ ) when K = 0; and φ ( ρ ) = sinh ρ, ρ ∈ [0 , ∞ ) when K = − , and dz is the induced standard metricon S n in Euclidean space. We denoteΦ( ρ ) = Z ρ φ ( s ) ds and consider the vector field V = φ ( ρ ) ∂∂ρ . It is well known that V is a conformalkilling field. Let M n ⊂ N n +1 ( K ) be a closed hypersurface and ν be the outwardunit normal vector field. We call function u = h V, ν i to be the generalized supportfunction of the hypersurface.Let κ = h κ , · · · , κ n i be the vector of n principle curvatures of the hypersurface M and we denote the k -th elementary symmetric function of κ by σ k ( κ ) . For
Key words and phrases.
Alexandrov-Fenchel type inequalities, sphere, convexity.
MR(2010)Subject Classification space form N n +1 , there exists a notion of quermassintegrals, which can be ex-pressed as a family of curvature integrals using Cauchy-Cronfton formulas (e.g.,[17]). Definition 1.1.
Let X : M → N n +1 ( K ) be a closed hypersurface embedded into N n +1 ( K ) and σ k be the k -th elementary symmetric function of the second funda-mental form. Suppose Ω is the domain enclosed by M n in N n +1 ( K ) . The k -thquermassintegal A k is defined as follows: A − (Ω) = Vol(Ω) , A (Ω) = Z M dµ g , A (Ω) = Z M σ dµ g + nK Vol(Ω) , A k (Ω) = Z M σ k dµ g + K ( n − k + 1) k − A k − (Ω) , where ≤ k ≤ n. In the hyperbolic space H n +1 , there have been many interesting results. Bren-dle, Hung and Wang [5] used the inverse mean curvature flow to obtain Minkowskitype inequality with weighted factor (the relation between R M σ φ ′ and R Ω φ ′ dvol )for a compact mean convex hypersurface which is star-shaped with respect to theorigin. In the case of h -convexity, full range of quermassintegral inequalities wereobtained in [7, 20] using expanding and contracting types of flows. Very recently,the results in [20] for h -convex domains in H n +1 were reproved using flow (1.1)directly in [14] by establishing that h -convexity is preserved along flow (1.1). Thesharp relation between A and A was previously proved in [15] by a differentmethod. Andrews, Chen and Wei [3] replaced the h -convexity assumption withthe weaker assumption of positive sectional curvature to obtain the relation be-tween A k and A − for 0 ≤ k ≤ n − . In the general case of k -convexity, Brendle,Guan and Li [4] considered the following parabolic evolution equation of a smoothone parameter family of embedded hypersurfaces X ( · , t ) ⊂ N n +1 ( K ) , (1.1) X t = ( φ ′ ( ρ ) F − uc n,k ) ν, where X is the position vector and F := σ k +1 ( κ ) σ k ( κ ) . The motivation to study Brendle-Guan-Li’ flow is the Alexandrov-Fenchel type inequalities for quermassintegralsin space forms. If one establishes long time existence and convergence of the flow(1.1), the sharp Alexandrov-Fenchel inequalities for k -star-shaped domains in N n +1 ( K ) would follow. In the case K = − , the main problem is the preservationof star-shapedness along the flow. As a consequence, Brendle, Guan and Li [4]established sharp inequalities between A n − and A k for convex domains in H n +1 , ∀ k ≤ n − . For general 0 ≤ k ≤ l ≤ n − H n +1 , with an extra initial gradientIN CHEN AND JUN SUN 3bound Condition (max x ∈ S n |∇ ln(cosh ρ ) | ( x, ≤
12 + 3 min t =0 sinh ρ ), Brendle,Guan and Li [4] can obtain the sharp inequalities of the relations between A k and A l .In S n +1 , Gir˜ ao and Pinheiro [9] used the inverse mean curvature flow to provethe Minkowski type inequality. Brendle, Guan and Li [4] established sharp in-equalities between A n − and A k for convex domains in S n +1 , ∀ k ≤ n − . Weiand Xiong proved the following optimal inequalities for convex hypersurfaces insphere.
Theorem 1.2. (see Theorem 1.3 in [21] ) Let Σ n be a closed and strictly convexhypersurface in S n +1 . Then we have the following optimal inequalities ( k ≤ n ) Z Σ L k dµ ≥ C kn (2 k )! ω nk n | Σ | n − kn . Equality holds if and only if Σ is a geodesic sphere. Here, the Gauss-Bonnet curvature is L k = C kn (2 k )! k X i =0 C ik C k − in ( − i σ k − i . Makowski and Scheuer proved the following Alexandrov-Fenchel type inequal-ities in the spheres. Namely
Theorem 1.3. (see Theorem 7.6 in [16] ) Let M ⊂ S n +1 be an embedded, closedconnected and convex C -hypersurface of the sphere. Let k ∈ N + with k + 1 ≤ n and let ˆ M be convex body enclosed by M . Then we have the inequality W k +1 ( ˆ M ) ≥ ω n n + 1 k X i =0 ( − i n − kn − k + 2 i (cid:18) ki (cid:19) ( n + 1 ω n W ( ˆ M )) n − k +2 in , and equality holds if and only if M is a geodesic sphere. By Proposition 7 in [18], we have(1.2) W k +1 = 1( n + 1) (cid:18) nk (cid:19) A k . In view of (1 . , Theorem 1.3 implies the relation between A k and A in S n +1 (Ge,Wang and Wu also proved a similar type of inequalities in H n +1 in Theorem 1.3 in[7] before). The optimal inequalities between two quermasssintegrals A k and A l (for general 0 ≤ k ≤ l ≤ n −
1) in S n +1 are still open. We will derive the relationbetween two adjacent quermassintegrals for general even number (i.e., A k and A k − ). We will also derive the relation between two adjacent quermassintegralsfor general odd number (i.e., A k +1 and A k − ) as well.Assume that s k = A k ( B π ( o )) , we now state the main result of this paper: Theorem 1.4.
Let M be a closed convex C -hypersurface in S n +1+ , then the fol-lowing inequality holds, (1.3) A k (Ω) ≥ ξ k,k − ( A k − (Ω)) for any ≤ k ≤ n − , where ξ k,k − is the unique positive function defined on (0 , s k − ) such that theequality holds if and only if M is a geodesic sphere. We will prove the relations between A k and A k − by induction. It is al-ready known that hypersurfaces converge smoothly to the equator along the Ger-hadt’s flow (see flow (4.1)). In Section 4, under the assumption that A k − (Ω) ≥ ξ k − ,k − ( A k − (Ω)) (the equality holds if and only if M is a geodesic sphere), wecould apply Gerhardt’s flow to prove that A k (Ω) ≥ ξ k,k − ( A k − (Ω))(the equalityholds if and only if M is a geodesic sphere). We can also use Gerhardt’s flowto derive the relation between A and A as well. However, we are unable touse Gerhardt’s flow to obtain the relation between A and A − as we are failedto get enough information about ξ , − as we describe in (4.15) (see also (4.9))by the definition of ξ k,k − . To derive the relation of A and A − , we will useChen-Guan-Li-Scheuer’s flow (see flow (3.1)) in Section 3. A nice feature of thistype of flow is that we do not need to know specific information about ξ , − andwe just need to obtain the monotonicity property of A and A − along the flowto acquire the relation between A and A − . More generally, we can prove therelation between A m and A − for 0 ≤ m ≤ n −
1. Except in the case of k = 0(i.e., flow (3.2)) where C estimates follow directly from the theory of quasi-linearPDE, C estimates for solutions of flow (3 .
1) is still an open question. As a result,combining applying Gerhardt’s flow with Chen-Guan-Li-Scheuer’s flow together,we could establish a full range of Alexandrov-Fenchel inequalities described inTheorem 1.4.The subsequent sections of this paper are organized as follows: in Section 2,we will recall some general facts on the elementary symmetric functions and thequermassintegrals; in Section 3, we will use Chen-Guan-Li-Scheuer’s flow to provethe the relation between A m and A − ; in Section 4, we will use Gerhardt’s flowto finish the proof of the main theorem.2. Setting and general facts
Let us present some basic facts which will be used later in this paper.
Definition 2.1. ( [10] ) For ≤ k ≤ n, let Γ k be a cone in R n determined by Γ k = { λ ∈ R n : σ > , · · · , σ k > } . An n × n symmetric matrix W is called belonging to Γ k if λ ( W ) ∈ Γ k . Then we will introduce Newton-Maclaurin inequalities and Minkowski identity.IN CHEN AND JUN SUN 5
Lemma 2.2. ( [10] ) For W ∈ Γ k , ( n − k + 1)( k + 1) σ k − ( W ) σ k +1 ( W ) ≤ k ( n − k ) σ k ( W ) , and σ k +1 ( W ) ≤ c n,k σ k +1 k k ( W ) , where c n,k = σ k +1 kk σ k +1 ( I ) . The equality holds if and only if W = cI for some c > . Proposition 2.3. ( [4] ) Let M be a closed hypersurfaces in N n +1 ( K ) . Then, for k = 0 , , · · · , n − , ( k + 1) Z M σ k +1 u = ( n − k ) Z M φ ′ ( ρ ) σ k ( κ ) . We will use the convention that σ ≡ . We define(2.1) Φ( ρ ) = Z ρ φ ( r ) dr. Then Φ( ρ ) is ρ , cosh ρ − , − cos ρ for K = 0 , − Lemma 2.4. ( [12] ) Let M n ⊂ N n +1 be a closed hypersurface with induced metric g . Let Φ be defined as in (2.1) and V = φ ( ρ ) ∂∂ρ . Then Φ | M satisfies, ∇ i ∇ j Φ = φ ′ ( ρ ) g ij − h ij h V, ν i , where ∇ is the covariant derivative with respect to g, ν is the outward unit normal,and h ij is the second fundamental form of the hypersurface. We also recall the gradient and hessian of the support function u := h V, ν i under the induced metric g on M . Lemma 2.5. ( [12] ) The support function u satisfies ∇ i u = h il ∇ l Φ; ∇ i ∇ j u = ∇ l h ij ∇ l Φ + φ ′ h ij − u ( h ) ij , where ( h ) ij := g kl h ik h jl . Next we will present the evolution equations of σ l and quermassintegrals. Let M ( t ) be a smooth family of closed hyersurfaces in N n +1 ( K ) . Let X ( · , t ) denote apoint on M ( t ) . We will consider the flow(2.2) X t = f ν. Along this flow, we have
Proposition 2.6. ( [4] ) Under the flow (2.2) for the hypersurface in a Riemann-ian manifold, suppose Ω is the domain enclosed by the closed hypersurface, wehave the following evolution equations. ∂ t g ij = 2 f h ij ∂ t h ij = −∇ i ∇ j f + f ( h ) ij − f R νijν ∂ t h ji = − g jk ∇ k ∇ j f − g jk f ( h ) ki − f g jk R νikν ∂ t σ k = ∂σ k ∂h ji ∂ t h ji Moreover, if N has constant sectional curvature K , then for l ≥ , we have (2.3) ∂ t Z M σ l = Z M f [( l + 1) σ l +1 − ( n − l + 1) Kσ l − ] dµ g , and ∂ t Vol(Ω) = Z M f dµ g . Using Proposition 2.6, we have the following proposition, which motivates thedefinition of the quermassintegrals:
Proposition 2.7. ( [4] ) In N n +1 ( K ) , along the flow (2.2) for ≤ l < n − , wehave (2.4) ∂ t A l = ( l + 1) Z M f σ l +1 . Let B ρ ( o ) ⊂ N n +1 ( K ) be the geodesic ball of radius ρ centered at the origin o .Then we have ddρ ( A k ( B ρ ( o ))) = ( k + 1) Z ∂B ρ ( o ) σ k +1 > , for any k = − , , , , · · · , n −
1. If we view A k ( B ρ ( o )) as a function of ρ , thenthe inverse function can be denoted as ρ = η k ( A k ( B ρ ( o ))) , where η k : (0 , s k ) → (0 , π/
2) is a strictly increasing function for any fixed k .Brendle, Guan and Li [4] want to compare the relation between A k and A l forgiven k > l and balls B ρ ( o ) , ρ > N n +1 ( K ) , which are the optimal solutionsof this isoperimetric problem. Let ξ k,l be the unique positive strictly increasingfunction defined on (0 , s l ) such that(2.5) A k ( B ρ ( o )) = ξ k,l ( A l ( B ρ ( o ))) . In general, for a bounded domain Ω ⊂ N n +1 ( K ) , they want to establish(2.6) A k (Ω) ≥ ξ k,l ( A l (Ω)) . IN CHEN AND JUN SUN 7In this following part of this paper, we will prove the relation between A k and A k − . More precisely, we will show that A k (Ω) ≥ ξ k,k − ( A k − (Ω)) . Chen-Guan-Li-Scheuer’s flow and applications
In this section, we will establish the relation between A (Ω) and A − (Ω). Ac-tually, we will provide the relationship between A m and A − for 1 ≤ m ≤ n − . In [6], Chen, Guan, Li and Scheuer introduced the following flow(3.1) X t = ( c n,k φ ′ − σ k +1 σ k u ) ν, Similar to the Brendle-Guan-Li’s flow, monotonicity property for quermassinte-grals holds as long as the flow (3.1) exists. It is claimed in [13] and proved in [6]that flow (3.1) preserves convexity, and one has C -estimates for solutions, andupper and lower bounds for F = σ k +1 σ k along the flow (3.1). For completeness, wewill include a proof of convexity preserving which we will use later here.3.1. Convexity preserving.
In this subsection, we will prove that if M isstrictly convex, then along the normalized flow (3.1), the principle curvaturesof M t remain strictly positive. We will need the following algebraic lemma. Lemma 3.1. ( [19] ) Let F = σ k +1 σ k ( h ij ) , and { ˜ h ij } be the inverse matrix of h ij , then ( F ij,rs + 2 F ir ˜ h js ) η ij η rs ≥ F − ( F ij η ij ) , for any real symmetric n × n matrix ( η ij ) . Here, F ij = ∂F∂h ij and F ij,rs = ∂ F∂h ij ∂h rs . Lemma 3.2.
Let X ( · , t ) be a smooth, closed and strictly convex solution to thenormalized flow (3.1) for t ∈ [0 , T ) , which encloses the origin. There is a positiveconstant C depending only on M , the upper and lower bounds of ρ ( C estimate),the gradient estimate ρ ( C estimate), and the upper bound of F such that theprincipal curvatures of X ( · , t ) are bounded from below κ i ( · , t ) ≥ C , ∀ t ∈ [0 , T ) and i = 1 , · · · , n. Proof.
From Proposition 2.6 and using the fact that ∇ i ∇ j φ ′ = − K ∇ i ∇ j Φ, wecompute ∂ t h ij = −∇ i ∇ j ( c n,k φ ′ − uF ) + ( c n,k φ ′ − uF )( h ) ij − K ( c n,k φ ′ − uF ) δ ij = − ( c n,k ∇ i ∇ j φ ′ − ∇ i ∇ j uF − ∇ i u ∇ j F − ∇ i F ∇ j u − u ∇ i ∇ j F )+ ( c n,k φ ′ − uF )( h ) ij − K ( c n,k φ ′ − uF ) δ ij = Kc n,k ∇ i ∇ j Φ + ∇ i ∇ j uF + ∇ i u ∇ j F + ∇ i F ∇ j u + u ∇ i ∇ j F − ( c n,k φ ′ − uF )( h ) ij − K ( c n,k φ ′ − uF ) δ ij = Kc n,k ( φ ′ g ij − uh ij ) + ( ∇ h ij ∇ Φ + φ ′ h ij − u ( h ) ij ) F + h il ∇ l Φ ∇ j F + h jl ∇ l Φ ∇ i F + u (cid:0) F kl ∇ k ∇ l h ij + ( F kl ( h ) kl − KF kk ) h ij − F (( h ) ij − Kg ij ) + F kl,pq ∇ i h pq ∇ j h kl (cid:1) + ( c n,k φ ′ − uF )( h ) ij − K ( c n,k φ ′ − uF ) δ ij . Thus ∂ t h ij = uF kl ∇ k ∇ l h ij + uF kl,pq ∇ i h pq ∇ j h kl + ∇ h ij ∇ Φ F + h il ∇ l Φ ∇ j F + h jl ∇ l Φ ∇ i F +( c n,k φ ′ − uF )( h ) ij + (cid:0) ( F kl ( h ) kl − KF kk ) u − c n,k Ku + φ ′ F (cid:1) h ij + 2 KuF δ ij . Assume (˜ h ij ) to be the inverse matrix of ( h ij ). The principle radii of curvature of M t are the eigenvalues of { ˜ h ik g kj } . To derive a positive lower bound of principlecurvatures, it suffices to prove that the eigenvalues of (˜ h ik g kj ) are bounded fromabove. For this, we consider the following quantity W ( x, t ) = log Λ( x, t ) − log u ( x, t ) , where Λ( x, t ) = max { ˜ h ij ( x, t ) ξ i ξ j : g ij ( x, t ) ξ i ξ j = 1 } . Assume W attains its maximum on S n × [0 , T ′ ] at ( x , t ) for any fixed T ′ < T . Wechoose a local orthonormal frame e , e , · · · , e n on M t such that ( h ij ) is diagonalat F ( x , t ). By a rotation, we may also suppose that Λ( x , t ) = ˜ h ij ( x, t ) ξ i ξ j with ξ = (1 , , · · · , . Let ω ( x, t ) = log λ ( x, t ) − log u ( x, t ) , where λ ( x, t ) = ˜ h /g . Then max S n × [0 ,T ′ ] W = max S n × [0 ,T ′ ] ω and so ω achievesits maximum at ( x , t ) . In the following we prove an upper bound for ω.∂ t λ = − (˜ h ) ∂ t h + ˜ h ∂ t g = − (˜ h ) (cid:16) uF kl ∇ k ∇ l h + uF kl,pq ∇ h pq ∇ h kl + ∇ h ∇ Φ F + h l ∇ l Φ ∇ F + h l ∇ l Φ ∇ F + ( c n,k φ ′ − uF )( h ) + (cid:0) ( F kl ( h ) kl − KF kk ) u − c n,k Ku + φ ′ F (cid:1) h + 2 KuF (cid:17) + 2( c n,k φ ′ − uF )= − (˜ h ) (cid:0) uF kl ∇ k ∇ l h + uF kl,pq ∇ h pq ∇ h kl (cid:1) + ∇ λ ∇ Φ F − h ∇ Φ ∇ F − ( c n,k φ ′ − uF ) − ˜ h (cid:0) ( F kl ( h ) kl − KF kk ) u − c n,k Ku + φ ′ F (cid:1) − KuF (˜ h ) + 2( c n,k φ ′ − uF )= uF kl ∇ k ∇ l λ − uF kl (˜ h ) ˜ h pq ∇ h kp ∇ h lq − u (˜ h ) F kl,pq ∇ h pq ∇ h kl + ∇ λ ∇ Φ F − h ∇ Φ ∇ F + ( c n,k φ ′ + uF ) − ˜ h (cid:0) ( F kl ( h ) kl − KF kk ) u − c n,k Ku + φ ′ F (cid:1) − KuF (˜ h ) , IN CHEN AND JUN SUN 9where we used the fact that ∇ i λ = − (˜ h ) h ,i , ∇ i ∇ j λ = − (˜ h ) ∇ i ∇ j h + 2˜ h pq ∇ h ip ∇ h jq . By Lemma 3.1, we have ∂ t λ ≤ uF kl ∇ k ∇ l λ − uF − (˜ h ) ( ∇ F ) + ∇ λ ∇ Φ F − h ∇ Φ ∇ F + ( c n,k φ ′ + uF ) − ˜ h (cid:0) ( F kl ( h ) kl − KF kk ) u − c n,k Ku + φ ′ F (cid:1) − KuF (˜ h ) . On the other hand, ∂ t u − uF kl u kl = f φ ′ − ∇ Φ ∇ f − uF kl u kl =( c n,k φ ′ − uF ) φ ′ − ∇ Φ ∇ ( c n,k φ ′ − uF ) − uF kl ( ∇ h kl ∇ Φ + φ ′ h kl − u ( h ) kl )= c n,k ( φ ′ ) + Kc n,k |∇ φ ′ | + ∇ Φ ∇ uF − uφ ′ F + u F kl ( h ) kl . Note that, at ( x , t ), ∇ i log λ = ∇ i log u, ∀ i = 1 , · · · , n, and ∇ i ∇ j (log λ − log u ) ≤ . Then we have1 λu ( u F kl ∇ k ∇ l λ − uλF kl ∇ k ∇ l u ) = uF kl ∇ k ∇ l (log λ − log u ) , λu ( uF ∇ λ ∇ Φ − λF ∇ Φ ∇ u ) = 0 . Thus0 ≤ ∂ t (log λ − log u ) = λ t λ − u t u ≤ λu (cid:16) u F kl ∇ k ∇ l λ − u F − (˜ h ) ( ∇ F ) + uF ∇ λ ∇ Φ − u ˜ h ∇ Φ ∇ F + u ( c n,k φ ′ + uF ) − ˜ h (cid:0) ( F kl ( h ) kl − KF kk ) u − c n,k Ku + φ ′ uF (cid:1) − Ku F (˜ h ) − (cid:0) λc n,k ( φ ′ ) + λKc n,k |∇ φ ′ | + ∇ Φ ∇ uλF − uλφ ′ F + λu F kl ( h ) kl ) + uλF kl ∇ k ∇ l u (cid:1)(cid:17) = uF kl ∇ k ∇ l (log λ − log u ) + 1 λu (cid:16)(cid:0) − u F − (˜ h ) ( ∇ F ) − u ˜ h ∇ Φ ∇ F (cid:1) − Ku (˜ h ) − ˜ h (cid:0) (2 F kl ( h ) kl − KF kk ) u − c n,k Ku − φ ′ uF − c n,k ( φ ′ ) + Kc n,k |∇ φ ′ | (cid:1) + u ( c n,k φ ′ + uF ) (cid:17) uF kl ∇ k ∇ l (log λ − log u ) + 1 λu (cid:16) F ( ∇ Φ) − Ku (˜ h ) − ˜ h (cid:0) (2 F kl ( h ) kl − KF kk ) u − c n,k Ku − φ ′ uF − c n,k ( φ ′ ) + Kc n,k |∇ φ ′ | (cid:1) + u ( c n,k φ ′ + uF ) (cid:17) . Now we assume K = 1 . It implies that − u (˜ h ) + ˜ h ( F kk u + c n,k ( u + ( φ ′ ) − |∇ φ ′ | ) + uφ ′ F )+ u ( c n,k φ ′ + uF ) + 12 F ( ∇ Φ) ≥ . Then we have (˜ h ) − c ˜ h − c ≤ , where the positive constants c , c depend on the upper and lower bound of ρ, thelower bound of u and the upper bound of F . Since u = φ √ φ + |∇ ρ | ( see equation(4.1) in [12]), we can conclude that the lower bound of u depends on the gradientestimate of ρ ( C estimate).The upper bound for ˜ h implies an upper bound for ω since u is bounded frombelow by a positive constant. This finishes the proof of the lemma. (cid:3) The relation between A m and A − . Now we will use the flow (3.1) toobtain the relation between A m and A − . In particular, the case m = 1 will beserved as the initial condition when we prove Theorem 1.4 by induction for oddnumbers. Proposition 3.3.
Let M be a closed convex C -hypersurface in S n +1+ , then thefollowing inequality holds, A m (Ω) ≥ ξ m, − ( A − (Ω)) , ∀ ≤ m ≤ n − , where ξ m, − is the unique positive function defined on (0 , s − ) such that the equal-ity holds if and only if M is a geodesic sphere.Proof. First of all, we can assume that the hypersurface is smooth and strictlyconvex, since otherwise we can use convolutions as in the proof of Corollary 1.2 in[16] to obtain a sequence of approximating smooth strictly convex hypersurfacesconverging in C to M . The inequality follows from the approximation. We willtreat the equality case A k (Ω) = ξ k,k − ( A k − (Ω)) for general 1 ≤ k ≤ n − k = 0, F = H and (3.1) becomes(3.2) X t = ( nφ ′ − uH ) ν. We have the C estimate (see Proposition 4.1 in [12]), C estimate(see Proposition5.2 in [12]) and the upper bound of H (see Corollary 3.3 in [12]). Then we see thatthe convexity is preserved by Lemma 3.2. Moreover, C estimate follows directlyIN CHEN AND JUN SUN 11from the theory of quasi-linear PDE. The surfaces converge exponentially to asphere as t → ∞ in the C ∞ topology by Theorem 1.1 in [12]. Along the flow(3.1), we have ddt A − = Z M ( t ) ( nφ ′ − Hu ) = 0 , and ddt A m = ( m + 1) Z M ( t ) ( nφ ′ − Hu ) σ m +1 dµ ≤ ( m + 1) Z M ( t ) ( nφ ′ σ m +1 − n ( m + 2) n − ( m + 1) uσ m +2 ) dµ = 0 , where we have used the Newton-Maclaurin inequality (Lemma 2.2) and Minkowskiidentity (Proposition 2.3). Then we have A m (Ω) ≥ ξ m, − ( A − (Ω)) for m = 1 , , · · · , n − . the equality holds only if M is a geodesic sphere. By the definition of ξ m, − , weknow that the equality holds if M is a geodesic sphere. Especially, A (Ω) ≥ ξ , − ( A − (Ω)) , with the equality holds if and if M is a geodesic sphere. (cid:3) Gerhardt’s flow and applications
In this section, following [4], we will use Gerhardt’s flow to prove the maintheorem. Gerhardt [8] considered the inverse curvature flows of strictly convexhypersurfaces in S n +1 and obtained smooth convergence of the flows to the equa-tor.Assume F = σ k σ k − . Then under the inverse curvature flow, X t = νF (4.1)with M (0) = ∂B ρ ( o ) , the geodesic spheres stay as geodesic spheres and for time t > M ( t ) = ∂B ρ ( t ) ( o ) . We first establish a Minkowski type inequality in S n +1 without weighted factor. Theorem 4.1.
Let M be a closed convex C -hypersurface in S n +1+ , then the fol-lowing inequality holds, ( Z M σ dµ g ) ≥ ξ ( A (Ω)) , where ξ is the unique positive function defined on (0 , s ) such that the equalityholds if and only if M is a geodesic sphere. Moreover, ξ satisfies that (4.2) 2 n − n ξ ( s ) = (2 n + 2 ξ ′ ( s )) s for s ∈ (0 , s ) . In fact we can express ξ explicitly by (4.3) ξ ( s ) = n ( n + 1) n ω n n +1 s n − n − n s, where ω n +1 is the volume of unit ball in R n +1 .Proof. Again we assume M to be smooth and strictly convex and we use thesame method as we used in the proof of Proposition 3.3.The function ξ is defined using the following relation:(4.4) ( Z ∂B ρ ( t ) ( o ) σ dµ g ) = ξ ( A ( B ρ ( t ) ( o ))) . Set f = H . Using (2.3) for l = 1 and (2.4) for l = 0, we have ddt (cid:0) ( Z ∂B ρ ( t ) ( o ) σ dµ g ) − ξ ( A ( B ρ ( t ) ( o ))) (cid:1) = 2 Z ∂B ρ ( t ) ( o ) σ dµ g Z ∂B ρ ( t ) ( o ) σ (2 σ − nσ ) dµ g − ξ ′ ( A ( B ρ ( t ) ( o ))) A ( B ρ ( t ) ( o )) Z ∂B ρ ( t ) ( o ) σ σ dµ g = 2 n − n ( Z ∂B ρ ( t ) ( o ) σ dµ g ) − n Z ∂B ρ ( t ) ( o ) σ dµ g Z ∂B ρ ( t ) ( o ) σ dµ g − ξ ′ ( A ( B ρ ( t ) ( o ))) A ( B ρ ( t ) ( o ))= 2 n − n ( Z ∂B ρ ( t ) ( o ) σ dµ g ) − n A ( B ρ ( t ) ( o )) − ξ ′ ( A ( B ρ ( t ) ( o ))) A ( B ρ ( t ) ( o )) , where the last step follows from the fact that(4.5) A (Ω( t )) ≤ Z M ( t ) σ dµ g Z M ( t ) σ dµ g , with the inequality strict unless σ = H is constant on M ( t ). Hence we obtainthat the equality holds if and only if M ( t ) is a geodesic sphere.By (4.4), we have2 n − n ξ ( A ( B ρ ( t ) ( o ))) − n A ( B ρ ( t ) ( o )) − ξ ′ ( A ( B ρ ( t ) ( o ))) A ( B ρ ( t ) ( o )) = 0 . IN CHEN AND JUN SUN 13We can obtain that2 n − n ξ ( s ) = (2 n + 2 ξ ′ ( s )) s for s ∈ (0 , s ) . Let M ( t ) solve the inverse curvature flow equation X t = σ ν with initial condition M (0) = M. Denote A k ( t ) = A k (Ω t ), where Ω t is the domain enclosed by M ( t ),then we have ddt (cid:0) ( Z M ( t ) σ dµ g ) − ξ ( A ( t )) (cid:1) = 2 Z M ( t ) σ dµ g Z M ( t ) σ (2 σ − nσ ) dµ g − ξ ′ ( A ( t )) A ( t ) Z M ( t ) σ σ dµ g ≤ n − n ( Z M ( t ) σ dµ g ) − n Z M ( t ) σ dµ g Z M ( t ) σ dµ g − ξ ′ ( A ( t )) A ( t ) ≤ n − n ( Z M ( t ) σ dµ g ) − n A ( t ) − ξ ′ ( A ( t )) A ( t )= 2 n − n (cid:0) ( Z M ( t ) σ dµ g ) − ξ ( A ( t )) (cid:1) . We will have ddt (cid:16) e − n − n t (cid:0) ( Z M ( t ) σ dµ g ) − ξ ( A ( t ) (cid:1)(cid:17) ≤ . Denote Q ( t ) = e − n − n t (cid:0) ( Z M ( t ) σ dµ g ) − ξ ( A ( t )) (cid:1) , then ddt Q ( t ) ≤ . Thus Q ( t ) − Q (0) ≤ , for all t ∈ [0 , T ∗ ) . It is proved in Theorem 1.1 in ([8]) that the curvature flow converges to anequator, as t → T ∗ , and with | π − ρ | m, S n ≤ c m Θ ∀ t ∈ [ t δ , T ∗ ) , where Θ = arccos e t − T ∗ . It follows thatVol(Ω t ) → Vol( B π ) , Z M t dµ g → | Σ( B π ) | , Z M t σ k dµ g → ∀ ≤ k ≤ n − , as t → T ∗ . By the definition of A k , we have A k ( t ) → A k ( B π ( o )) ∀ − ≤ k ≤ n − , as t → T ∗ , which implies that lim t → T ∗ Q ( t ) = 0 . Therefore, we have ( Z M (0) σ dµ g ) − ξ ( A (0)) ≥ , i.e., ( Z M σ dµ g ) − ξ ( A (Ω)) ≥ , with the equality holds only if M is a geodesic sphere. By the definition of ξ , weknow that the equality holds if M is a geodesic sphere.Finally, we will derive the explicit expression for ξ . First we can solve the ODE(4.2) to obtain(4.6) ξ ( s ) = s n − n ǫ − n − n ξ ( ǫ ) − n ( s − ǫ n s n − n ) . It remains to compute the limit lim ǫ → ǫ − n − n ξ ( ǫ ). For this purpose, we noticethat since the metric on S n +1 is given by ds = dρ + sin ρdz , we see that(4.7) A ( B ρ ( o )) = | ∂B ρ ( o ) | = sin n ρ | S n | = ( n + 1) ω n +1 sin n ρ. On the other hand, for ∂B ρ (0) ⊂ S n +1 , we have κ = κ = · · · κ n = cos ρ sin ρ . Hence,(4.8) Z ∂B ρ ( o ) σ dµ g = n cos ρ sin ρ | ∂B ρ ( o ) | = n ( n + 1) ω n +1 cos ρ sin n − ρ. Therefore, if we choose s = A ( B ρ ( o )) = ( n + 1) ω n +1 sin n ρ , then ξ ( s ) = Z ∂B ρ ( o ) σ dµ g ! = n ( n + 1) ω n +1 cos ρ sin n − ρ = n ( n + 1) ω n +1 cos ρ (cid:18) s ( n + 1) ω n +1 (cid:19) n − n = n ( n + 1) n ω n n +1 cos ρs n − n . Since ρ → s →
0, we see thatlim ǫ → ǫ − n − n ξ ( ǫ ) = lim ρ → (cid:16) n ( n + 1) n ω n n +1 cos ρ (cid:17) = n ( n + 1) n ω n n +1 . Now (4.3) follows by letting ǫ → (cid:3) Now we will establish the relation between A and A . IN CHEN AND JUN SUN 15
Proposition 4.2.
Let M be a closed convex C -hypersurface in S n +1+ , then thefollowing inequality holds, A (Ω) ≥ ξ , ( A (Ω)) , where ξ , is the unique positive function defined on (0 , s ) such that the equalityholds if and only if M is a geodesic sphere. Moreover, ξ , satisfies that (4.9) ξ ′ , ( s ) = ( n − ξ , ( s ) − ( n − sns . In fact we can express ξ , explicitly by (4.10) ξ , ( s ) = n ( n − n + 1) n ω n n +1 s n − n − n − s, where ω n +1 is the volume of unit ball in R n +1 .Proof. Again we assume M to be smooth and strictly convex and we use thesame method as we used in the proof of Proposition 3.3.For 0 < ρ ( t ) < π , the function ξ , is defined by(4.11) A ( B ρ ( t ) ( o )) − ξ , ( A ( B ρ ( t ) ( o ))) = 0 . Using (2.4) for l = 0 and l = 2 and considering the flow (4.1) with F = σ σ , wehave ddt (cid:0) A ( B ρ ( t ) ( o )) − ξ , ( A ( B ρ ( t ) ( o )) (cid:1) = 3 Z ∂B ρ ( t ) ( o ) σ σ σ dµ g − ξ ′ , ( A ( B ρ ( t ) ( o ))) Z ∂B ρ ( t ) ( o ) σ σ σ dµ g = 2( n − n − Z ∂B ρ ( t ) ( o ) σ dµ g − nn − ξ ′ , ( A ( B ρ ( t ) ( o ))) Z ∂B ρ ( t ) ( o ) σ dµ g , where the last step follows from the fact that the geodesic sphere is totally umbilic.(4.11) yields that(4.12) 2( n − n − Z ∂B ρ ( t ) ( o ) σ dµ g = 2 nn − ξ ′ , ( A ( B ρ ( t ) ( o ))) Z ∂B ρ ( t ) ( o ) σ dµ g . From the definition of A k , we have(4.13) A ( B ρ ( t ) ( o )) = Z ∂B ρ ( t ) ( o ) σ dµ g + K ( n − − A ( B ρ ( t ) ( o )) . Combining (4.11), (4.12) and (4.13), we have ξ ′ , ( s ) = ( n − ξ , ( s ) − ( n − s ) ns . M ( t ) solve the inverse curvature flow equation X t = σ σ ν with initial condition M (0) = M. Denote A k ( t ) = A k (Ω t ), where Ω t is the domain enclosed by M ( t ),then we have ddt (cid:0) A ( t ) − ξ , ( A ( t )) (cid:1) = 3 Z M ( t ) σ σ σ dµ g − ξ ′ , ( A ( t )) Z M ( t ) σ σ σ dµ g ≤ n − n − Z M ( t ) σ dµ g − nn − ξ ′ , ( A ( t )) Z M ( t ) σ dµ g = 2( n − n − A ( t ) − K ( n − − A ( t )) − nn − ξ ′ , ( A ( t )) A ( t )= 2( n − n − A ( t ) − K ( n − − A ( t )) − nn − n − ξ , ( A ( t )) − ( n − A ( t ) n A ( t ) A ( t )= 2( n − n − A ( t ) − ξ , ( A ( t ))) . We have ddt ( e − n − n − t ( A ( t ) − ξ , ( A ( t )))) ≤ . Denote Q ( t ) = e − n − n − t ( A ( t ) − ξ , ( A ( t ))) , then ddt Q ( t ) ≤ . Thus Q ( t ) − Q (0) ≤ , for all t ∈ [0 , T ∗ ) . By the definition of A k , we have lim t → T ∗ Q ( t ) = 0 . Therefore, we have A (0) − ξ , ( A (0)) ≥ , i.e., A (Ω) ≥ ξ , ( A (Ω)) , the equality holds only if M is a geodesic sphere. By the definition of ξ , , weknow that the equality holds if M is a geodesic sphere.IN CHEN AND JUN SUN 17Finally, we will derive the explicit expression for ξ , . First we can solve theODE (4.9) to obtain(4.14) ξ , ( s ) = s n − n ǫ − n − n ξ , ( ǫ ) − n −
12 ( s − ǫ n s n − n ) . It remains to compute the limit lim ǫ → ǫ − n − n ξ , ( ǫ ). For this purpose, recall thatthe area of the geodesic sphere is given by (4.7). On the other hand, for ∂B ρ (0) ⊂ S n +1 , we have κ = κ = · · · κ n = cos ρ sin ρ . Hence, we have σ = n ( n − ρ sin ρ so that A ( B ρ ( o )) = Z ∂B ρ ( o ) σ dµ g + ( n − A ( B ρ ( o ))= n ( n − ρ sin ρ | ∂B ρ ( o ) | + ( n − | ∂B ρ ( o ) | . Therefore, if we choose s = A ( B ρ ( o )) = | ∂B ρ ( o ) | = ( n + 1) ω n +1 sin n ρ , then ξ , ( s ) = A ( B ρ ( o )) = n ( n − ρ sin ρ s + ( n − s = n ( n − ρs (cid:18) s ( n + 1) ω n +1 (cid:19) − n + ( n − s = n ( n − n + 1) n ω n n +1 cos ρs n − n + ( n − s. Since ρ → s →
0, we see thatlim ǫ → ǫ − n − n ξ , ( ǫ ) = lim ρ → (cid:18) n ( n − n + 1) n ω n n +1 cos ρ + ( n − ǫ n (cid:19) = n ( n − n + 1) n ω n n +1 . Now (4.10) follows by letting ǫ → (cid:3) Remark . We can compare Theorem 4.1 and Proposition 4.2 with Theorem1.5 (see (1.5) and (1.6)) of [16]. The method we used to deduce Theorem 4.1and Proposition 4.2 can be used to derive the relation between A k and A k − forgeneral 1 ≤ k ≤ n − . Next, we will study the properties of the function ξ k,k − , which will be used inthe proof of the main theorem. Proposition 4.3.
For any s ∈ (0 , s k − ) , the following holds (4.15) ξ ′ k,k − ( s ) = n − kn − k + 2 ξ k,k − ( s ) − K n − k +1 k − ss − K n − k +3 k − ξ − k − ,k − ( s ) for k ≥ , where ξ k,k − and ξ k − ,k − are defined as in (2.5). Proof.
For k ≥ < ρ ( t ) < π , by the definition of ξ k,k − , we have(4.16) A k ( B ρ ( t ) ( o )) − ξ k,k − ( A k − ( B ρ ( t ) ( o ))) = 0 . By (2.4), we have along the flow (4.1) with F = σ k σ k − ddt (cid:0) A k ( B ρ ( t ) ( o )) − ξ k,k − ( A k − ( B ρ ( t ) ( o )) (cid:1) = ( k + 1) Z ∂B ρ ( t ) ( o ) σ k +1 σ k σ k − dµ g − ( k − ξ ′ k,k − ( A k − ( B ρ ( t ) ( o ))) Z ∂B ρ ( t ) ( o ) σ k − σ k σ k − dµ g = k ( n − k ) n − k + 1 Z ∂B ρ ( t ) ( o ) σ k dµ g − ( n − k + 2) kn − k + 1 ξ ′ k,k − ( A k − ( B ρ ( t ) ( o ))) Z ∂B ρ ( t ) ( o ) σ k − dµ g , where the last step follows from the fact that the geodesic sphere is totally umbilic.(4.16) yields that k ( n − k ) n − k + 1 Z ∂B ρ ( t ) ( o ) σ k dµ g = ( n − k + 2) kn − k + 1 ξ ′ k,k − ( A k − ( B ρ ( t ) ( o ))) Z ∂B ρ ( t ) ( o ) σ k − dµ g . From the definition of A k , we have A k ( B ρ ( t ) ( o )) = Z ∂B ρ ( t ) ( o ) σ k dµ g + K n − k + 1 k − A k − ( B ρ ( t ) ( o )) A k − ( B ρ ( t ) ( o )) = Z ∂B ρ ( t ) ( o ) σ k − dµ g + K n − k + 3 k − A k − ( B ρ ( t ) ( o ))Under the assumption that A k − ( B ρ ( o )) = ξ k − ,k − ( A k − ( B ρ ( o ))) , we have k ( n − k ) n − k + 1 (cid:0) ξ k,k − ( A k − ( B ρ ( t ) ( o )) (cid:1) − K n − k + 1 k − A k − ( B ρ ( t ) ( o )) (cid:1) − ( n − k + 2) kn − k + 1 (cid:0) A k − ( B ρ ( t ) ( o )) − K ( n − k + 3) k − ξ − k − ,k − ( A k − ( B ρ ( t ) ( o ))) (cid:1) · ξ ′ k,k − ( A k − ( B ρ ( t ) ( o ))) = 0 . Then we have ξ ′ k,k − ( s ) = n − kn − k + 2 ξ k,k − ( s ) − K n − k +1 k − ss − K n − k +3 k − ξ − k − ,k − ( s )IN CHEN AND JUN SUN 19for any s ∈ (0 , s k − ) . (cid:3) Now we can prove our main result.
Proof of Theorem 1.4.
Case 1. M is a closed strictly convex and smooth hyper-surface. We will prove Theorem 1.4 by reduction.Let M ( t ) solve the inverse curvature flow equation X t = σ k − σ k ν with initialcondition M (0) = M. Denote A k ( t ) = A k (Ω t ), where Ω t is the domain enclosedby M ( t ). For k = 1 ,
2, the inequality (1.3) holds by Proposition 3.3 with m = 1and Proposition 4.2, respectively. Then we can assume that A k − (Ω) ≥ ξ k − ,k − ( A k − (Ω)) , for any strictly convex hypersurface M in S n +1 and the equality holds if and onlyif M is a geodesic sphere. Since ddt ( A k − ( t )) = ( k − Z M ( t ) σ k − σ k σ k − dµ g ≥ k ( n − k + 2) n − k + 1 Z M ( t ) σ k − dµ g > M ( t ) converges to an equator of S n +1 , we have A k − ( t ) ∈ (0 , s k ) . By Proposition 4.3 and applying Newton-Maclaurin inequality, we have ddt (cid:0) A k ( t ) − ξ k,k − ( A k − ( t )) (cid:1) =( k + 1) Z M ( t ) σ k +1 σ k σ k − dµ g − ( k − ξ ′ k,k − ( A k − ( t )) Z M ( t ) σ k − σ k σ k − dµ g ≤ k ( n − k ) n − k + 1 Z M ( t ) σ k dµ g − ( n − k + 2) kn − k + 1 ξ ′ k,k − ( A k − ( t )) Z M ( t ) σ k − dµ g = k ( n − k ) n − k + 1 ( A k ( t ) − K n − k + 1 k − A k − ( t )) − ( n − k + 2) kn − k + 1 ξ ′ k,k − ( A k − ( t ))( A k − ( t ) − K n − k + 3 k − A k − ( t ))= k ( n − k ) n − k + 1 ( A k ( t ) − K n − k + 1 k − A k − ( t )) − ( n − k + 2) kn − k + 1 n − kn − k + 2 ξ k,k − ( A k − ( t )) − K n − k +1 k − A k − ( t ) A k − ( t ) − K n − k +3 k − ξ − k − ,k − ( A k − ( t )) · ( A k − ( t ) − K n − k + 3 k − A k − ( t )) ≤ k ( n − k ) n − k + 1 ( A k ( t ) − K n − k + 1 k − A k − ( t )) − k ( n − k ) n − k + 1 (cid:18) ξ k,k − ( A k − ( t )) − K n − k + 1 k − A k − ( t ) (cid:19) , A k − ( t ) ≤ ξ − k − ,k − ( A k − ( t )) . Therefore we have ddt (cid:0) A k ( t ) − ξ k,k − ( A k − ( t ) (cid:1) ≤ k ( n − k ) n − k + 1 ( A k ( t ) − ξ k,k − ( A k − ( t ))) . Assume that Q k ( t ) = e − k ( n − k ) n − k +1 t ( A k ( t ) − ξ k,k − ( A k − ( t ))) , then ddt Q k ( t ) ≤ . Thus Q k ( t ) − Q k (0) ≤ , for all t ∈ [0 , T ∗ ) . By the definition of A k , we have lim t → T ∗ Q k ( t ) = 0 . Therefore, A k (0) − ξ k,k − ( A k − (0)) ≥ , i.e., A k (Ω) ≥ ξ k,k − ( A k − (Ω)) , the equality holds only if M is a geodesic sphere. By the definition of ξ k,k − , weknow that the equality holds if M is a geodesic sphere.Case 2. M is a closed convex C -hypersurface.We can obtain a sequence of approximating smooth strictly convex hypersur-faces converging in C to M . The inequality follows from the approximation. Wenow treat the equality case A k (Ω) = ξ k,k − ( A k − (Ω)) for general 1 ≤ k ≤ n − M is convex and the equality holds for general 1 ≤ k ≤ n −
1, wewill show M is strictly k -convex. To see this, note that both A k and A k − arepositive, since there exists at least one elliptic point on a closed hypersurface in S n +1+ . Let M + = { x ∈ M | σ k ( x ) > } . M + is open and nonempty. We claim that M + is closed. This would imply M = M + , so M is strictly k convex.We now prove that M + is closed. We will follow the idea of [11]. Pick any η ∈ C ( M + ) compactly supported in M + . Let M s be the hypersurface determinedby position function X s = X + sην, where X is the position function of M and ν is the unit outernormal of M at X . Let Ω s be the domain enclosed by M s . Itis easy to show that M s is k -convex when s is small enough. Define I k (Ω s ) = A k (Ω s ) − ξ k,k − ( A k − (Ω s )) . Therefore I k (Ω s ) − I k (Ω) ≥ s small, which implies that dds I k (Ω s ) | s =0 = 0 . IN CHEN AND JUN SUN 21Simple calculation yields ∂ t A l = ( l + 1) Z M f σ l +1 . Therefore, dds I k (Ω s ) | s =0 = ( k + 1) Z M ( σ k +1 − c σ k − ) ηdµ g = 0 , where c = k − k +1 ξ ′ k,k − ( A k − ) > η ∈ C ( M + ) . Thus,(4.17) σ k +1 = c σ k − , ∀ x ∈ M + . It follows from the Newton-Maclaurine inequality that there is a dimensionalconstant C ( n, k ) such that σ k +1 ≤ C ( n, k ) σ k − k − ( x ) , x ∈ M + . By (4.17), there is a positive c , such that σ k − ≥ c > , ∀ x ∈ M + , where c = ( c C ( n,k ) ) k − is a positive constant depending only on n, k and Ω . Inview of (4.17), we have σ k +1 ≥ c c , ∀ x ∈ M + , which implies that σ k ≥ c > , where c = q ( n − k +1)( k +1) k ( n − k ) c c is a positive constant depending only on n, k andΩ . . It follows that M + is closed.Then we claim that the flow (3 .
1) preserves the convexity in a short time.We denote M by M . Approximate the initial surface M by a strictly convexones M ǫ , by the implicit function theorem, there exists a t > ǫ ) such that (3 .
1) has a regular solution M ǫ ( t ) (with C , C , C bounds) whichsatisfies that M ǫ (0) = M ǫ for 0 ≤ t ≤ t . Strict convexity is preserved for theapproximate flows M ǫ ( t ) by Lemma 3.2. Letting ǫ → , we could obtain that M ( t ) is convex for any 0 ≤ t ≤ t .Now we will prove that the equality A k (Ω t ) = ξ k,k − ( A k − (Ω t )) would be pre-served at least in a short time along the flow (3 . ddt A k = ( k + 1) Z ( c n,k φ ′ σ k +1 − σ k +1 σ k ) ≤ , and ddt A k − = ( k − Z ( c n,k φ ′ σ k − − σ k +1 σ k − σ k ) ≥ . A k (Ω t ) ≤ A k (Ω) and A k − (Ω) ≤ A k − (Ω t ) for any 0 ≤ t ≤ t .Since A k (Ω) = ξ k,k − ( A k − (Ω)) , we have A k (Ω t ) ≤ A k (Ω) = ξ k,k − ( A k − (Ω)) ≤ ξ k,k − ( A k − (Ω t )) , along the flow (3 . A k (Ω t ) ≥ ξ k,k − ( A k − (Ω t )).As a result, the equality of the Newton-Maclaurin inequality must be held atevery point of M(t) for each 0 ≤ t ≤ t . This implies that M ( t ) is a geodesicsphere for each 0 ≤ t ≤ t . In particular, M is a geodesic sphere.This finishes the proof of the theorem. (cid:3) Acknowledgements.
The authors would like to thank Professor Jiayu Liand Professor Pengfei Guan for helpful suggestions. The authors would alsobe grateful for useful discussions with Professor Yong Wei.
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Min Chen, University of Science and Technology of China, No.96, JinZhaiRoad Baohe District,Hefei,Anhui, 230026,P.R.China.
Email address : [email protected] Jun Sun, School of Mathematics and Statistics, Wuhan University, Wuhan,and Hubei Key Laboratory of Computational Science (Wuhan University), Wuhan,430072, P. R. of China.
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