A fully-nonlinear flow and quermassintegral inequalities in the sphere
aa r X i v : . [ m a t h . A P ] F e b A FULLY-NONLINEAR FLOW AND QUERMASSINTEGRALINEQUALITIES IN THE SPHERE
CHUANQIANG CHEN, PENGFEI GUAN, JUNFANG LI, AND JULIAN SCHEUER
Dedicated to Joseph Kohn on the occasion of his 90th birthday
Abstract.
This expository paper presents the current knowledge of particular fullynonlinear curvature flows with local forcing term, so-called locally constrained curvatureflows. We focus on the spherical ambient space. The flows are designed to preserve aquermassintegral and to de-/increase the other quermassintegrals. The convergence ofthis flow to a round sphere would settle the full set of quermassintegral inequalities forconvex domains of the sphere, but a full proof is still missing. Here we collect what isknown and hope to attract wide attention to this interesting problem. Introduction
Let M n be a smooth, closed and connected manifold and let X : M n ֒ → S n +1 be theembedding of a strictly convex hypersurface. Let p ∈ S n +1 be a point in the interior of theconvex body enclosed by M , such that M lies in the interior of the hemisphere determinedby p and denote by ds = dρ + φ ( ρ ) dz (1.1)the metric in polar coordinates around p , where φ ( ρ ) = sin( ρ ), ρ ∈ [0 , π ), is the radialdistance, and dz is the induced standard metric on S n .We consider the following locally constrained curvature flow in the sphere: ∂X∂t = ( c n,k φ ′ ( ρ ) − u σ k +1 ( λ ) σ k ( λ ) ) ν, (1.2) X ( · ,
0) = X ( · ) . where X ( x, t ) ∈ S n +1 is the position vector of the evolving hypersurface M ( t ), ν theoutward unit normal, u = h φ ( ρ ) ∂∂ρ , ν i , λ = ( λ , · · · , λ n ) the principal curvatures, X : M ֒ → S n +1 the initial embedded hypersurface, σ k the k -th elementary symmetric function,and c n,k = σ k +1 ( I ) σ k ( I ) = n − kk +1 , I = (1 , · · · , Mathematics Subject Classification.
Key words and phrases.
Fully-nonlinear flow, Quermassintegral inequalities, Constant rank theorem.Research of CC was supported by NSFC NO. 11771396, research of PG was supported in part by NSERCDiscovery Grant, research of JL was supported in part by NSF DMS-1007223, and JS was supported bythe ”Deutsche Forschungsgemeinschaft” (DFG, German research foundation), Project ”Quermassintegralpreserving local curvature flows”, No. SCHE 1879/3-1.
The particular interest in these flows stems from its monotonicity properties with respectto the quermassintegrals for convex bodies Ω in the sphere. Let M = ∂ Ω, set A − =Vol(Ω) , A = Z M dµ g , A = Z M σ ( λ ) dµ g + n Vol(Ω) , (1.3) A m = Z M σ m ( λ ) dµ g + n − m + 1 m − A m − , where 2 ≤ m ≤ n . Here g is the induced metric on M and dµ g the associated volumeelement.The monotonicity properties of those functionals along the flow (1.2) follow from thefollowing Hsiung-Minkowski identities (see Proposition 2.6):( m + 1) Z M uσ m +1 ( λ ) = ( n − m ) Z M φ ( ρ ) ′ σ m ( λ ) , ≤ m ≤ n − . (1.4)With the help of the evolution equations (see Proposition 2.9): ∂ t A − = Z M (cid:18) c n,k φ ′ ( ρ ) − u σ k +1 ( λ ) σ k ( λ ) (cid:19) dµ g , (1.5)and ∂ t A l = ( l + 1) Z M σ l +1 ( λ ) (cid:18) c n,k φ ′ ( ρ ) − u σ k +1 ( λ ) σ k ( λ ) (cid:19) dµ g , (1.6)we deduce that along the flow (1.2) for 0 ≤ k ≤ n −
1, the following monotonicity relationshold: ∂ t A l ≥ , if l < k − , if l = k − ≤ , if l > k − . (1.7)Hence, if one can prove that the flow (1.2) moves an arbitrary convex hypersurface to around sphere, then the following conjecture would turn into a theorem: Conjecture 1.1. A l ≤ ξ l,k ( A k ) , ∀ − ≤ l < k ≤ n, (1.8)where ξ l,k is the unique positive function defined on (0 , ∞ ) such that “ = ” holds when M is a geodesic sphere. “ = ” holds if and only if M is a geodesic sphere.Flow (1.2) is another example of hypersurface flows which have been introduced recentlywith goals to establish optimal geometric inequalities [7, 3, 8, 9, 11] for hypersurfaces inspace forms. These locally constrained flows are associated to the optimal solutions tothe problems of calculus of variations in geometric setting. The counterpart of (1.2) in R n +1 was considered in [8, 9], where the longtime existence and convergence were provedby transforming the equation to corresponding inverse type PDE on S n for the supportfunction. In the case of S n +1 , up to several special values of k and l , this conjecture isopen until today, see for example [3, 4, 13]. The main issue is that so far we can not FULLY-NONLINEAR FLOW AND QUERMASSINTEGRAL INEQUALITIES 3 control the curvature along the flow (1.2) from above (except the case k = 0 [8]). All theother a priori estimates for this flow are in place and this note is supposed to collect thoseestimates.The rest of this article is organized as follows. In section 2, we list some basic facts for k -th elementary symmetric functions, hypersurfaces in S n +1 and evolution equations. Insection 3, we prove the C , C a priori estimates and uniform bounds of F = σ k +1 ( λ ) σ k ( λ ) . Insection 4, we prove the strict convexity of M ( t ) along the flow (1.2) if M is convex. Inthe last section, we give a discussion of the C estimate.2. Preliminary
We first recall some well-known facts about k -th elementary symmetric functions, hy-persurfaces in S n +1 , and then give some evolution equations along the flow (1.2).2.1. Elementary symmetric functions.
For any k = 1 , · · · , n , and λ = ( λ , · · · , λ n ),the k -th elementary symmetric function is defined as follows(2.1) σ k ( λ ) = X ≤ i
Let λ = ( λ , . . . , λ n ) ∈ R n and k = 1 , · · · , n , then σ k ( λ ) = σ k ( λ | i ) + λ i σ k − ( λ | i ) , ∀ ≤ i ≤ n, X i λ i σ k − ( λ | i ) = kσ k ( λ ) , X i σ k ( λ | i ) = ( n − k ) σ k ( λ ) . Viewing σ k as a function on symmetric matrices, we also denote by σ k ( W | i ) the sym-metric function with W deleting the i -row and i -column and σ k ( W | ij ) the symmetricfunction with W deleting the i, j -rows and i, j -columns. Then we have the followingidentities. Proposition 2.2.
Suppose W = ( W ij ) is diagonal, and m is a positive integer, then ∂σ m ( W ) ∂W ij = ( σ m − ( W | i ) , if i = j, , if i = j. and ∂ σ m ( W ) ∂W ij ∂W kl = σ m − ( W | ik ) , if i = j, k = l, i = k, − σ m − ( W | ik ) , if i = l, j = k, i = j, , otherwise . CHUANQIANG CHEN, PENGFEI GUAN, JUNFANG LI, AND JULIAN SCHEUER
Recall that the G˚arding’s cone is defined as(2.2) Γ k = { λ ∈ R n : σ i ( λ ) > , ∀ ≤ i ≤ k } . The following properties are well known.
Proposition 2.3.
Let λ ∈ Γ k and k ∈ { , , · · · , n } . Suppose that λ ≥ · · · ≥ λ k ≥ · · · ≥ λ n , then we have σ k − ( λ | n ) ≥ σ k − ( λ | n − ≥ · · · ≥ σ k − ( λ | k ) ≥ · · · ≥ σ k − ( λ | > λ ≥ · · · ≥ λ k > , σ k ( λ ) ≤ C kn λ · · · λ k ;(2.4) σ k ( λ ) ≥ λ · · · λ k , if λ ∈ Γ k +1 ;(2.5) where C kn = n ! k !( n − k )! . The generalized Newton-MacLaurin inequality is as follows, which will be used all thetime. See [17].
Proposition 2.4.
For λ ∈ Γ k and k > l ≥ , r > s ≥ , k ≥ r , l ≥ s , we have " σ k ( λ ) /C kn σ l ( λ ) /C ln k − l ≤ " σ r ( λ ) /C rn σ s ( λ ) /C sn r − s . (2.6)2.2. Hypersurfaces in S n +1 . The following lemma is well known, e.g. [8].
Lemma 2.5.
Let M n ⊂ S n +1 be a closed hypersurface with induced metric g . Let Φ =Φ( ρ ) = R ρ φ ( r ) dr , then ∇ i ∇ j Φ = φ ′ ( ρ ) g ij − h ij u, (2.7) recall u = h φ ( ρ ) ∂∂ρ , ν i . We have the following Hsiung-Minkowski identities, see [8].
Proposition 2.6.
Let M be a closed hypersurface in S n +1 . Then, for m = 0 , , · · · , n − , ( m + 1) Z M uσ m +1 ( λ ) = ( n − m ) Z M φ ′ ( ρ ) σ m ( λ ) , (2.8) where we use the convention that σ = 1 . Next, we state the gradient and hessian of the support function u = h φ ( ρ ) ∂∂ρ , ν i underthe induced metric g on M , see [8]. Lemma 2.7.
The support function u satisfies ∇ i u = g ml h im ∇ l Φ , (2.9) ∇ i ∇ j u = g ml ∇ m h ij ∇ l Φ + φ ′ h ij − ( h ) ij u, (2.10) where ( h ) ij = g ml h im h jl . FULLY-NONLINEAR FLOW AND QUERMASSINTEGRAL INEQUALITIES 5
Evolution equations.
Let M ( t ) be a smooth family of closed hypersurfaces in S n +1 ,and X ( · , t ) denote a point on M ( t ). The following basic evolution equations for normalvariations are well known, e.g. [6]. Proposition 2.8.
Under the flow ∂ t X = f ( X ( · , t )) ν in the sphere we have the followingevolution equations ∂ t g ij =2 f h ij , (2.11) ∂ t dµ g = f σ ( λ ) dµ g , (2.12) ∂ t h ij = − ∇ i ∇ j f + f ( h ) ij − f g ij , (2.13) ∂ t h ij = − ∇ i ∇ j f − f g im ( h ) mj − f δ ij . (2.14)From Proposition 2.8, we can obtain the evolution of the quermassintegrals in the sphere. Proposition 2.9.
Along the flow ∂ t X = f ( X ( · , t )) ν in the sphere, we have for ≤ l ≤ n − ∂ t A l = ( l + 1) Z M σ l +1 ( λ ) f dµ g , (2.15) and ∂ t A − = Z M f dµ g , (2.16) where Ω is the domain enclosed by the closed hypersurface. Moreover, if the flow is (1.2) and M ( t ) is strictly convex, then we have ∂ t A l ≥ , if l < k − , if l = k − ≤ , if l > k − . (2.17) Proof. (2.15) and (2.16) follow directly form Proposition 2.8. (2.17) follows from (2.8),(2.15) and the Newton-MacLaurin inequality. (cid:3)
Let (
M, g ) be a hypersurface in S n +1 with induced metric g . We now give the localexpressions of the induced metric, second fundamental form, Weingarten curvatures etc.when M is a graph of a smooth and positive function ρ ( z ) on S n . Let ∂ , · · · , ∂ n bea local frame along M and ∂ ρ be the vector field along the radial direction. Then thesupport function, induced metric, inverse metric matrix, second fundamental form canbe expressed as follows. For simplicity, all the covariant derivatives with respect to thestandard spherical metric e ij will also be denoted as ∇ when there is no confusion in the CHUANQIANG CHEN, PENGFEI GUAN, JUNFANG LI, AND JULIAN SCHEUER context. u = φ p φ + |∇ ρ | , (2.18) g ij = φ e ij + ρ i ρ j , (2.19) g ij = 1 φ ( e ij − ρ i ρ j φ + |∇ ρ | ) , (2.20) h ij = 1 p φ + |∇ ρ | ( − φ ∇ i ∇ j ρ + 2 φ ′ ρ i ρ j + φ φ ′ e ij ) , (2.21) h ij = 1 φ p φ + |∇ ρ | ( e im − ρ i ρ m φ + |∇ ρ | )( − φ ∇ m ∇ j ρ + 2 φ ′ ρ m ρ j + φ φ ′ e mj ) , (2.22)where all the covariant derivatives ∇ and ρ i are w.r.t. the spherical metric e ij .We now consider the flow equation (1.2) of radial graphs over S n in S n +1 . Let ω = φ p φ + |∇ ρ | . It is known that if a family of radial graphs satisfy ∂ t X = f ν , then the evolution of thescalar function ρ = ρ ( X ( z, t ) , t ) satisfies ∂ t ρ = f ω. (2.23)The following is a well known commutator identity. Lemma 2.10.
Let h ij be the second fundamental form and g ij be the induced metric of ahypersurface in S n +1 . Then ∇ i ∇ j h ml = ∇ m ∇ l h ij + h ij ( h ) ml − ( h ) ij h ml + h il ( h ) mj − ( h ) il h mj + [ h ml g ij − h ij g ml + h mj g il − h il g mj ] . (2.24) Lemma 2.11.
Along the flow (1.2) in S n +1 , the graph function ρ and the support function u = h φ ( ρ ) ∂∂ρ , ν i evolve as follows (2.25) ∂ t ρ − uF ij ∇ i ∇ j ρ = φ ′ φ u ( c n,k − F ij g ij ) + φ ′ φ uF ij ρ i ρ j ∂ t u − uF ij ∇ i ∇ j u = − c n,k ∇ Φ ∇ φ ′ + F ∇ Φ ∇ u + ( c n,k φ ′ − uF ) φ ′ + u F ij ( h ) ij , (2.26) where F = σ k +1 ( λ ) σ k ( λ ) and F ij = ∂F∂h ij .Proof. The function ρ satisfies ∂ t ρ = ( c n,k φ ′ − uF ) ω, and uF ij ρ ij = − uωF + φ ′ φ uF ij g ij − φ ′ φ uF ij ρ i ρ j . Hence (2.25) holds.
FULLY-NONLINEAR FLOW AND QUERMASSINTEGRAL INEQUALITIES 7
Applying Lemma 2.7, we can obtain ∂ t u − uF ij u ij = f φ ′ − ∇ Φ ∇ f − uF ij [ ∇ h ij ∇ Φ + φ ′ h ij − ( h ) ij u ]= ( c n,k φ ′ − uF ) φ ′ − ∇ Φ ∇ ( c n,k φ ′ − uF ) − u [ ∇ F ∇ Φ + φ ′ F − F ij ( h ) ij u ]= − c n,k ∇ Φ ∇ φ ′ + F ∇ Φ ∇ u + ( c n,k φ ′ − uF ) φ ′ + u F ij ( h ) ij . (cid:3) Lemma 2.12.
Let h ij be the second fundamental form and g ij be the induced metric of ahypersurface in S n +1 and F = σ k +1 ( λ ) σ k ( λ ) . Then ∇ i ∇ j F = F αβ ∇ α ∇ β h ij + F αβ,γη ∇ i h αβ ∇ j h γη + [ F αβ ( h ) αβ − F αα ] h ij − F [( h ) ij − g ij ] , (2.27) where F αβ = ∂F∂h αβ and F αβ,γη = ∂ F∂h αβ ∂h γη .Proof. ∇ i ∇ j F = F αβ ∇ i ∇ j h αβ + F αβ,γη ∇ i h αβ ∇ j h γη = F αβ ∇ α ∇ β h ij + F αβ,γη ∇ i h αβ ∇ j h γη + F αβ [ h ij ( h ) αβ − ( h ) ij h αβ + h iβ ( h ) αj − ( h ) iβ h αj + ( h αβ g ij − h ij g αβ + h iβ g αj − h αj g iβ )]= F αβ ∇ α ∇ β h ij + F αβ,γη ∇ i h αβ ∇ j h γη + [ F αβ ( h ) αβ − F αα ] h ij − F [( h ) ij − g ij ] . (2.28) (cid:3) Lemma 2.13.
Let h ij be the second fundamental form and g ij be the induced metric of ahypersurface in S n +1 and F = σ k +1 ( λ ) σ k ( λ ) . Then along the flow (1.2) ∂ t h ij − uF ml ∇ m ∇ l h ij = uF ml,pq ∇ i h ml ∇ j h pq + ∇ i u ∇ j F + ∇ j u ∇ i F + F ∇ h ij ∇ Φ − ( c n,k φ ′ + uF )( h ) ij + h ij [ u ( F ml ( h ) ml − F mm ) + φ ′ F − c n,k u ]+ 2 uF δ ij , (2.29) and ∂ t F − uF ml ∇ m ∇ l F =2 F ml ∇ m u ∇ l F + F ∇ Φ ∇ F − [ c n,k F ml ( h ) ml − F ] φ ′ + uF [ X F mm − c n,k ] . (2.30) CHUANQIANG CHEN, PENGFEI GUAN, JUNFANG LI, AND JULIAN SCHEUER
Proof.
By the tensorial property, we do not distinguish upper and lower indexes in thisproof whenever applicable. We need the fact that ∇ φ ′ = −∇ Φ, and then we can obtain ∂ t h ij = − ∇ i ∇ j ( c n,k φ ′ − uF ) − ( c n,k φ ′ − uF )( h ) ij − ( c n,k φ ′ − uF ) δ ij = − c n,k ∇ i ∇ j φ ′ + u ∇ i ∇ j F + ∇ i u ∇ j F + ∇ j u ∇ i F + F ∇ i ∇ j u − ( c n,k φ ′ − uF )( h ) ij − ( c n,k φ ′ − uF ) δ ij = c n,k ( φ ′ g ij − uh ij ) + ∇ i u ∇ j F + ∇ j u ∇ i F + F [ ∇ h ij ∇ Φ + φ ′ h ij − u ( h ) ij ]+ u [ F ml ∇ m ∇ l h ij + F ml,pq ∇ i h ml ∇ j h pq + ( F ml ( h ) ml − F mm ) h ij − F (( h ) ij − g ij )] − ( c n,k φ ′ − uF )( h ) ij − ( c n,k φ ′ − uF ) δ ij = u [ F ml ∇ m ∇ l h ij + F ml,pq ∇ i h ml ∇ j h pq ] + ∇ i u ∇ j F + ∇ j u ∇ i F + F ∇ h ij ∇ Φ − ( c n,k φ ′ + uF )( h ) ij + h ij [ u ( F ml ( h ) ml − F mm ) + φ ′ F − c n,k u ]+ 2 uF δ ij . Finally, (2.30) follows from ∂ t F = F ij ∂ t h ji . (cid:3) Lemma 2.14.
Let h ij be the second fundamental form and g ij be the induced metric of ahypersurface in S n +1 and F = σ k +1 ( λ ) σ k ( λ ) . Then along the flow (1.2) ∂ t ( uF ) − uF ml ∇ m ∇ l ( uF ) = F [ c n,k |∇ Φ | + ( c n,k φ ′ − uF ) φ ′ + u F ml ( h ) ml ]+ u [ uF ( X F mm − c n,k ) − ( c n,k F ml ( h ) ml − F ) φ ′ ]+ F ∇ Φ ∇ ( uF ) , (2.31) and ∂ t ( h ii u ) − uF ml ∇ m ∇ l ( h ii u ) = F ml,pq ∇ i h ml ∇ i h pq + 2 u ∇ i u ∇ i F + F ∇ Φ ∇ ( h ii u )+ 2 F ml ∇ m u ∇ l ( h ii u ) − ( c n,k φ ′ u + F )( h ) ii + 2 F + h ii [ F mm − ( c n,k φ ′ u − Fu ) φ ′ − c n,k + c n,k u ∇ Φ ∇ φ ′ ] . (2.32) 3. A priori estimates
Since M is strictly convex, there is T > M ( t ) is strictly convex for all 0 ≤ t < T . This will be assumed in the rest of this section.
FULLY-NONLINEAR FLOW AND QUERMASSINTEGRAL INEQUALITIES 9 C estimate.Theorem 3.1. Let M be a strictly convex, radial graph of positive function ρ over S n embedded in S n +1 . If M ( t ) solves the flow (1.2) with the initial value M , then for any ( z, t ) ∈ S n × [0 , T ) min z ∈ S n ρ ( z ) ≤ ρ ( z, t ) ≤ max z ∈ S n ρ ( z ) . (3.1) Proof.
At critical points of ρ , we have the following critical point conditions, ∇ ρ = 0 , ω = 1 , u = φ, (3.2)and then the Weingarten curvature is h ij = 1 φ ( − ρ ij + φφ ′ e ij ) , (3.3)and then F = σ k +1 ( λ ) σ k ( λ ) = φ ′ φ σ k +1 ( − ρ ij φφ ′ + e ij ) σ k ( − ρ ij φφ ′ + e ij ) . (3.4)So at the critical point, we have ∂ t ρ = c n,k φ ′ − uF = c n,k φ ′ − φ ′ σ k +1 ( − ρ ij φφ ′ + e ij ) σ k ( − ρ ij φφ ′ + e ij ) . (3.5)By standard maximum principle, this proves the upper and lower bounds for ρ . (cid:3) C estimates.Lemma 3.2. Let F ( λ ) := σ k +1 ( λ ) σ k ( λ ) , and c n,k = F ( I ) = n − kk +1 where I = (1 , · · · , . Then X i F ii λ i ≥ F c n,k , X i F ii ≥ c n,k , ∀ λ ∈ Γ k . (3.6) Moreover, if λ ∈ ¯Γ k +1 , then P F ii ≤ n − k .Proof. The proof below is from [3] We first derive(3.7) P F ii λ i = P σ k ( λ | i ) λ i σ k − σ k +1 σ k − ( λ | i ) λ i σ k = [ σ σ k +1 − ( k +2) σ k +2 ] σ k − σ k +1 [ σ σ k − ( k +1) σ k +1 ] σ k = ( k +1) σ k +1 − ( k +2) σ k +2 σ k σ k ≥ k +1 n − k F = c n,k F , where the last inequality follows from Newton-McLaurin inequality. Similarly, we have(3.8) P F ii = P σ k ( λ | i ) σ k ( λ ) − σ k +1 ( λ ) σ k − ( λ | i ) σ k = ( n − k ) σ k − ( n − k +1) σ k +1 σ k − σ k ≥ n − kk +1 = c n,k , where the last inequality follows from Newton-McLaurin inequality. If λ ∈ ¯Γ k +1 , then σ k +1 σ k − ≥ P F ii ≤ n − k from the second identity in (3.8). (cid:3) Theorem 3.3.
Let M be a strictly convex, radial graph of positive function ρ over S n embedded in S n +1 . If M ( t ) solves the flow (1.2) with the initial value M , then for any ( z, t ) ∈ S n × [0 , T ) u ( z, t ) ≥ min z ∈ S n u ( z, . (3.9) As a consequence, we have C bound for ρ , that is, | ρ | C ( S n ) ≤ C, (3.10) where C depends only on the initial data.Proof. For any fixed t ∈ (0 , T ), we have at the minimum point of u ( z, t ), ∇ u = 0 . So from Lemma 2.11, we have ∂ t u − uF ij u ij = c n,k |∇ Φ | + ( c n,k φ ′ − uF ) φ ′ + u F ij ( h ) ij ≥ c n,k |∇ Φ | + c n,k ( φ ′ − c n,k uF ) ≥ , (3.11)which finishes the proof of (3.9). (cid:3) Uniform bounds of F .Lemma 3.4. Let λ = ( λ , · · · , λ n ) ∈ Γ n , and λ ≥ · · · ≥ λ n . Then for ≤ m ≤ n − ,we have m ( n − m ) σ m ( λ ) − ( m + 1)( n − m + 1) σ m +1 ( λ ) σ m − ( λ ) σ m ( λ ) ∼ ( λ − λ n ) λ , (3.12) where f ∼ g means C ( n,m ) g ≤ f ≤ C ( n, m ) g for some constant C ( n, m ) > .Proof. The case m = 1 is trivial, as( n − σ ( λ ) − nσ ( λ ) = X i We may assume 1 < < n . By direct computation we can derive m ( n − m ) σ m ( λ ) − ( m + 1)( n − m + 1) σ m +1 ( λ ) σ m − ( λ )=[ mσ m ( λ )][( n − m ) σ m ( λ )] − [( m + 1) σ m +1 ( λ )][( n − m + 1) σ m − ( λ )]=[ X i λ i σ m − ( λ | i )][ X j σ m ( λ | j )] − [ X j λ j σ m ( λ | j )][ X i σ m − ( λ | i )]= X i,j ( λ i − λ j ) σ m − ( λ | i ) σ m ( λ | j )= X i Let M be a strictly convex, radial graph of positive function ρ over S n embedded in S n +1 . If M ( t ) solves the flow (1.2) with the initial value M , then for any ( z, t ) ∈ S n × [0 , T ) 1 C ≤ F ≤ C, (3.15) where C depends only on n , k and the initial data.Proof. For any fixed t ∈ (0 , T ), we have at the critical points of F , ∇ F = 0 . So from Lemma 2.13, we can get ∂ t F − uF ij ∇ i ∇ j F = − c n,k φ ′ F [ F ij ( h ) ij F − c n,k ] + uF [ X F ii − c n,k ] . (3.16)In the following, we divide the proof of (3.15) into three cases.Firstly, for the Case: 1 ≤ k ≤ n − 2, we know from Lemma 3.4, F ij ( h ) ij F − c n,k = ( k + 1) n − k − n − k σ k +1 − ( k + 2) σ k +2 σ k σ k +1 ∼ ( λ − λ n ) λ , (3.17)and X F ii − c n,k = kk +1 ( n − k ) σ k − ( n − k + 1) σ k +1 σ k − σ k ∼ ( λ − λ n ) λ . (3.18)Thus F ij ( h ) ij F − c n,k ∼ X F ii − c n,k . (3.19)Hence (3.15) holds.Secondly, for the Case: k = 0, we can directly get, F ij ( h ) ij F − c n,k ≥ , (3.20)and X F ii − c n,k = 0 , (3.21)so we can get F ≤ C from (3.16). To prove the lower bound of F , we consider the minimumpoint of uF , and then we can get F ≥ C from (2.31). Hence (3.15) holds.Lastly, we consider the Case: k = n − F ij ( h ) ij F − c n,k = 0 , (3.22)and ( n − k ) − c n,k ≥ X F ii − c n,k ≥ , (3.23)so we can get F ≥ C from (3.16). FULLY-NONLINEAR FLOW AND QUERMASSINTEGRAL INEQUALITIES 13 To prove the upper bound of F , we consider the maximum point of P =: F + u . At themaximum point of P , we have 0 = ∇ i P = ∇ i F − u ∇ i u, (3.24)and then ∂ t P − uF ij ∇ i ∇ j P = ∂ t F − uF ij ∇ i ∇ j F − u (cid:2) ∂ t u − uF ij ∇ i ∇ j u (cid:3) − uF ij · u ∇ i u ∇ j u =2 F ij ∇ i u ∇ j F + F ∇ Φ ∇ F + uF [ X F ii − c n,k ] − u (cid:2) − c n,k |∇ Φ | + F ∇ Φ ∇ u + ( c n,k φ ′ − uF ) φ ′ + u F ij ( h ) ij (cid:3) − uF ij · u ∇ i u ∇ j u = uF [ X F ii − c n,k ] − u (cid:20) − c n,k |∇ Φ | + ( c n,k φ ′ − uF ) φ ′ + u F c n,k (cid:21) . (3.25)So we can get F ≤ C from (3.25). Hence (3.15) holds. (cid:3) Preserving convexity In this section, we prove the flow (1.2) preserves convexity in S n +1 . Denote T > T ∗ is the largest time of existence of a smooth solution to (1.2). Theorem 4.1. Let M ( t ) be an oriented immersed connected hypersurface in S n +1 with apositive semi-definite second fundamental form h ( t ) ∈ Γ k +1 satisfying equation (1.2) for t ∈ [0 , T ∗ ) , then then M ( t ) is strictly convex for all t ∈ (0 , T ∗ ) . Here we provide two proofs. The first proof follows from the following lemma. Lemma 4.2. Along the solution of (1.2) with a strictly convex initial hypersurface M ⊂ S n +1 all flow hypersurfaces M t = X ( M, t ) are strictly convex up to T ∗ , i.e. T = T ∗ , witha uniform estimate h ij ≥ cδ ij , where c = c (sup M ρ, inf M ρ, n, k, T ∗ ) .Proof. We calculate the evolution equation of the inverse { b ij } = { h ji } − , which is welldefined up to T . We suppose that T < T ∗ . ∂ t b mm = − b mr ∂ t h rs b sm ,F ij ∇ i ∇ j b mm =2 F ij b mr ∇ i h rs b sp ∇ j h pq b qm − F ij b mr ∇ i ∇ j h rs b sm , ∇ i u = h ji ∇ j Φ , and by evolution equation for h ji , we deduce ∂ t b mm − uF ij ∇ i ∇ j b mm − F ∇ i Φ ∇ i b mm = − u ( F pq,rs + 2 F qs b pr ) ∇ i h pq ∇ j h rs b mj b im − b mj ∇ j F ∇ m Φ − b im ∇ i F ∇ m Φ − uF ij h ir h rj b mm + ( c n,k φ ′ + uF ) − φ ′ F b mm (4.1) + uF ij g ij b mm + c n,k ub mm − uF b mr b rm ≤ − uF ∇ i F ∇ j F b mj b im − b mj ∇ j F ∇ m Φ − b im ∇ i F ∇ m Φ+ ψ ( t ) b mm + ψ ( t ) − uF b mr b rm , where we used the inverse concavity of F , cf. Theorem 2.3 in [1] and where ψ i are smoothfunctions which are uniformly bounded up to T , due to the uniform upper and lowerbounds of F .We use a well known trick to estimate the maximal eigenvalue of b , e.g. compareLemma 6.1 [5]. Let Q = sup { b ij η i η j | g ij η i η j = 1 } , and suppose this function attains a maximum at ( t , ξ ) with t < T, i.e. Q ( t , ξ ) = sup [0 ,t ] × M Q. Choose coordinates in ( t , ξ ) with g ij = δ ij , b ij = λ − i δ ij , λ − ≤ · · · ≤ λ − n . Let η be the vector field η = (0 , . . . , , 1) and define˜ Q = b ij η i η j g ij η i η j , then locally around ( t , ξ ) we have ˜ Q ≤ Q and the derivatives coincide. Thus at ( t , ξ )the function ˜ Q and b nn satisfy the same evolution equation and we may show that theright hand side of (4.1) is negative at the point ( t , ξ ) in these coordinates, yielding acontradiction.In these coordinates we obtain ∂ t b nn − uF ij ∇ i ∇ j b nn − F ∇ i Φ ∇ i b nn ≤ − uF ( ∇ n F ) λ − n − λ − n ∇ n F ∇ n Φ + ψ ( t ) λ − n + ψ ( t ) − uF λ − n ≤ − uF ( ∇ n F ) λ − n + ǫλ − n ( ∇ n F ) + c ǫ ( ∇ n Φ) + ψ ( t ) λ − n + ψ ( t ) − uF λ − n < , (4.2)for small ǫ and large λ − n . Hence we obtain that λ − n does not blow up at T , in contradictionto the definition of T < T ∗ . Hence we must have T = T ∗ , with a uniform lower bound on h ij on finite intervals. The proof is complete. (cid:3) FULLY-NONLINEAR FLOW AND QUERMASSINTEGRAL INEQUALITIES 15 The second proof is from the Constant Rank Theorem in Bian-Guan [2] , along the linesof proof of Theorem 6.1 in [8] (where the constant rank theorem was proved for a generalflow in R n +1 ). For (1.2) in S n +1 , we have an extra good term which is associated to thecurvature K = 1. We outline the arguments here with necessary modification. Proof. Let W = ( g im h mj ), and l ( t ) be the minimal rank of W . Suppose at W is degenerate( x , T ), such that W ( T ) attains minimal rank l < n at x . Set ϕ ( x, t ) = σ l +1 ( W ( x, t )) + σ l +2 ( W ( x, t )) σ l +1 ( W ( x, t )) . It is proved in section 2 in [2] that ϕ is in C , .As in Bian-Guan [2], near ( x , T ), the index set { , , · · · , n } can be divided in to twosubsets B, G , where for i ∈ B , the eigenvalues of { W ij } , λ i is small and for j ∈ G , λ j isstrictly positive away from 0. As in [2], we may assume at each point of computation, { W ij } is diagonal. Notice that W ii ≤ Cϕ for all i ∈ B .Denote G = uF − c n,k φ ′ and F = σ k +1 ( λ ) σ k ( λ ) . From (2.14) we recall ∂ t h ii = ∇ i ∇ i G + Gg ik ( h ) ki + G. (4.3)So we have the following equality X G αβ ϕ αβ − ϕ t = O ( ϕ + X i,j ∈ B |∇ W ij | ) − σ ( B ) X αβ X i = j ∈ B G αβ W ij,α W ij,β − σ ( B ) X αβ X i ∈ B G αβ ( W ii,α σ ( B ) − W ii X j ∈ B W jj,α )( W ii,β σ ( B ) − W ii X j ∈ B W jj,β ) − X i ∈ B [ σ l ( G ) + σ ( B | i ) − σ ( B | i ) σ ( B ) ] X αβ X j ∈ G G αβ W ij,α W ij,β W jj + X i ∈ B [ σ l ( G ) + σ ( B | i ) − σ ( B | i ) σ ( B ) ] X αβ G αβ W ii,αβ − ∂ t W ii (4.4)By (4.3) and (2.29), ∂ t h ii = u [ F αβ ∇ α ∇ β h ii + F αβ,γη ∇ i h αβ ∇ i h γη ] + O ( ϕ + X i,j ∈ B |∇ W ij | ) + 2 uF, (4.5)where we used the facts ∇ i u = O ( ϕ ), ∇ i ∇ i u = O ( ϕ + |∇ W ii | ) and ∇ i ∇ i Φ = φ ′ + O ( ϕ ), ∀ i ∈ B .As G αβ = uF αβ , X αβ G αβ W ii,αβ − ∂ t W ii = u X αβ F αβ ∇ α ∇ β h ii − ∂ t h ii = O ( ϕ + X i,j ∈ B |∇ W ij | ) − uF αβ,γη W αβ,i W γη,i − uF, (4.6) Since F satisfies the structure condition in [2] F αβ,γη W αβ,i W γη,i + 2 X αβ X j ∈ G F αβ W ij,α W ij,β W jj ≥ . We obtain X G αβ ϕ αβ − ϕ t ≤ C ( ϕ + |∇ ϕ | ) − C X i,j ∈ B |∇ W ij | − uF X i ∈ B [ σ l ( G ) + σ ( B | i ) − σ ( B | i ) σ ( B ) ] . Following the analysis in the proof of Theorem 3.2 in [2], it yields X G αβ ϕ αβ − ϕ t ≤ C ( ϕ + |∇ ϕ | ) − uF σ l ( G ) . This is a contradiction from the standard strong maximum principle for parabolic equa-tions. (cid:3) Remark . The preservation convexity of flow (1.2) when k = 0 was first observed byJS and Chao Xia [15].Since in the case of k = 0, the longtime existence and convergence was proved in [8],the following sharp inequality follows. Proposition 4.4. If Ω ⊂ S n +1 is convex, then Vol(Ω) ≤ ξ l, − ( A l ) ∀ l ≥ , with equality holds iff Ω is a convex geodesic ball. Discussion of C estimate The curvature estimate for flow (1.2) is still open. In the case of R n +1 , φ ′ = 1, thecorresponding flow is(5.1) X t = ( c n,k − uF ( λ )) ν. Flow (5.1) has the same curvature estimate issue. In [9, 10] the flow (5.1) was convertedto a corresponding inverse type flow for the Euclidean support function u of the evolvingconvex body parametrized on the outer normals (i.e. on S n ). The longtime existence andconvergence of the admissible solution were proved in [9, 10].Below we will convert flow (1.2) to an evolution of convex bodies in R n +1 and we writedown the evolution equation of the corresponding Euclidean support function ˜ u on S n .Introduce a new variable γ satisfying dγdρ = 1 φ . (5.2) FULLY-NONLINEAR FLOW AND QUERMASSINTEGRAL INEQUALITIES 17 Let ω = p |∇ γ | , one can compute the unit outward normal ν = ω (1 , − ∇ ρφ ), and u = φω , (5.3) g ij = φ e ij + ρ i ρ j , (5.4) g ij = 1 φ ( e ij − γ i γ j ω ) , (5.5) h ij = φω ( − γ ij + φ ′ γ i γ j + φ ′ e ij ) , (5.6) h ij = 1 φω ( e im − γ i γ m ω )( − γ mj + φ ′ γ m γ j + φ ′ e mj ) . (5.7)It follows from (2.23) that the evolution equation for γ is ∂ t γ = 1 φ ∂ t ρ = f ωφ = c n,k φ ′ u − σ k +1 ( λ ) σ k ( λ ) . (5.8)From (5.7), h ij = 1 φω ( e im − γ i γ m ω )( − γ mj + φ ′ γ m γ j + φ ′ e mj )= 1 φω (cid:8) ( e im − γ i γ m ω )( − γ mj + γ m γ j + e mj ) + ( φ ′ − δ ij (cid:9) = e γ φ e h ij + φ ′ − φω δ ij , (5.9)where e h ij = 1 e γ ω (cid:8) ( e im − γ i γ m ω )( − γ mj + γ m γ j + e mj ) , which is the Weingarten tensor of the graph f M (over S n ) of radial function e ρ = e γ in R n +1 .Since M ( t ) is strictly convex (i.e. { h ij } > φ ′ − ρ − < 0, thus { e h ij } > f M is strictly convex.We have ∂ t e ρ = e ρ∂ t γ = c n,k φ ′ φ e ρω − e ρ φ σ k +1 σ k ( e h ij + φ ′ − e ρω δ ij ) . (5.10)Let e u be the support function of the strictly convex body f M , and z and ν be the unit radialvector and the unit outer normal vector of f M , respectively. Then from e ρ ( z, t )( z · ν ) = e u ( ν, t ), we can get log e ρ ( z, t ) = log e u ( ν, t ) − log( z · ν ) and1 e ρ ( z, t ) ∂ e ρ ( z, t ) ∂t = 1 e u ( ν, t ) [ ∇ e u · ν t + e u t ] − z · ν t z · ν = 1 e u ( ν, t ) ∂ e u ( ν, t ) ∂t + 1 e u ( ν, t ) [( ∇ e u − e ρ ( z, t ) z ) · ν t ]= 1 e u ( ν, t ) ∂ e u ( ν, t ) ∂t . Denote W e u =: { e u ij + e uδ ij } = { e h ij } − and we can get the evolution equation of e u as follows ∂ t e u = e u e ρ ∂ t e ρ = c n,k φ ′ φ e uω − e ρ e uφ σ k +1 σ k ( e h ij + φ ′ − e ρω δ ij )= : G ( W e u , e u, ∇ e u ) . (5.11)This equation is of inverse type, and the question is whether (5.11) exists for all time? Acknowledgments: Part of this work was done while CC was visiting McGill University in 2018, he wouldlike to thank McGill and PG for the warm hospitality.This work was made possible through a research scholarship JS received from the DFGand which was carried out at Columbia University in New York. JS would like to thankthe DFG, Columbia University and especially Prof. Simon Brendle for their support.Parts of this work were written during a visit of JS to McGill University in Montreal.JS would like to thank McGill and PG for their hospitality and support. References [1] B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions , J. Reine Angew.Math. 608 (2007), 17-33.[2] B. Bian and P. Guan, A microscopic convexity principle for nonlinear partial differential equations ,Invent. Math. 177 (2009), no. 2, 307-335.[3] S. Brendle, P. Guan and J. Li, An inverse curvature type hypersurface flow in Space forms , preprint,2014.[4] Min Chen and Jun Sun, Alexandrov-Fenchel type inequalities in the sphere , arXiv:2101.09419, 2021.[5] C. Gerhardt, Closed Weingarten hypersurfaces in space forms, Geometric analysis and the calculus ofvariations (Jurgen Jost, ed.) , International Press of Boston Inc., 1996, pp. 71–98.[6] C. 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Sinestrari, Convexity estimates for mean curvature flow and singularities of meanconvex surfaces , Acta Math. 183 (1999), no. 1, 45-70.[13] M. Makowski and J. Scheuer, Rigidity results, inverse curvature flows and Alexandrov-Fenchel typeinequalities in the sphere , Asian J. Math. 20 (2016), no. 5, 869-892.[14] J. Scheuer, Isotropic functions revisited , Arch. Math. 110 (2018), no. 6, 591-604.[15] J. Scheuer and C. Xia, Locally constrained inverse curvature flows , Trans. Amer. Math. Soc. 372(2019), no. 10, 6771-6803.[16] F. Schulze, Nonlinear evolution by mean curvature and isoperimetric inequalities , J. Differential Geom.,79 (2008), no. 2, 197-241.[17] J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations , Clay MathematicsProceedings, volume 2, 2005, 283-309. Chuanqiang Chen, School of Mathematics and Statistics, Ningbo University, Ningbo,315211, Zhejiang Province, P.R. China Email address : [email protected] Pengfei Guan, Department of Mathematics, McGill University, Montreal, Quebec,H3A2K6, Canada Email address : [email protected] Junfang Li, Department of Mathematics, University of Alabama at Birmingham, Birming-ham, AL 35294, USA Email address : [email protected] Julian Scheuer, School of Mathematics, Cardiff University, Cardiff CF24 4AG, Wales,UK Email address ::