Flow by Gauss curvature to Dual Orlicz-Minkowski problems
aa r X i v : . [ m a t h . A P ] J a n FLOW BY GAUSS CURVATURE TO DUAL ORLICZ-MINKOWSKI PROBLEMS
LI CHEN, QIANG TU, DI WU, NI XIANGA bstract . In this paper we study a normalised anisotropic Gauss curvature flow of strictly convex, closedhypersurfaces in the Euclidean space R n + . We prove that the flow exists for all time and convergessmoothly to the unique, strictly convex solution of a Monge-Amp`ere type equation. Our argument pro-vides a parabolic proof in the smooth category for the existence of solutions to the Dual Orlicz-Minkowskiproblem introduced by Zhu, Xing and Ye. Keywords : Gauss curvature flow, convex hypersurface, Monge-Amp`ere equation.
MSC : Primary 53C44, Secondary 35K96.
1. I ntroduction
As we known, the Gauss curvature flow was introduced by Firey [14] to model the shape changeof worn stones. The first celebrated result was proved by Andrews in [3] for Gauss curvature flow,where Firey’s conjecture that convex surfaces moving by their Gauss curvature become spherical as theycontract to points was proved. Guan and Ni [15] proved that convex hypersurfaces in R n + contractingby the Gauss curvature flow converge (after rescaling to fixed volume) to a smooth uniformly convexself-similar solution of the flow. Soon, Andrews, Guan and Ni [7] extended the results in [15] to theflow by powers of the Gauss curvature K α with α > n + . Recently, Brendle, Choi and Daskalopoulos[11] proved that round spheres are the only closed, strictly convex self-similar solutions to the K α -flowwith α > n + . Therefore, the generalized Firey’s conjecture proposed by Andrews in [6] was completelysolved, that is, the solutions of the flow by powers of the Gauss curvature converge to spheres for any α > n + . We also refer to [12, 1, 4, 5] and the references therein.As a natural extension of Gauss curvature flows, anisotropic Gauss curvature flows have attractedconsiderable attention and they provide alternative proofs for the existence of solutions to elliptic PDEsarising in geometry and physics, especially for the Minkowski-type problem. For example a alternativeproof based on the logarithmic Gauss curvature flow was given by Chou-Wang in [13] for the classicalMinkowski problem, in [21] for a prescribing Gauss curvature problem. Using a contracting Gausscurvature flow, Li-Sheng-Wang [17] have provided a parabolic proof in the smooth category for theclassical Aleksandrov and dual Minkowski problems. Recently, two kinds of normalised anisotropicGauss curvature flow are used to prove the L p dual Minkowski problems by Chen-Huang-Zhao [9] andChen-Li [10], respectively. These results are major source of inspiration for us.Let M be a smooth, closed, strictly convex hypersurface in R n + enclosing the origin. In this paper,we study the long-time behavior of the following normalised anisotropic Gauss curvature flow which isa family of hypersurfaces M t given by smooth maps X : M × [0 , T ) → R n + satisfying the initial valueproblem(1.1) ∂ X ∂ t = − θ ( t ) f ( ν ) r n + ϕ ( r ) K ν + X , X ( · , = X , This research was supported by Hubei Provincial Department of Education Key Projects D20171004, D20181003 and theNational Natural Science Foundation of China No.11971157. where ν is the unit outer vector of M t at X , K denotes the Gauss curvature of M t at X , r = | X | denotesthe distance form X to the origin, f ∈ C ∞ ( S n ) with f >
0, and θ ( t ) = Z S n ϕ ( r ( ξ, t )) d ξ (cid:20) Z S n f ( x ) dx (cid:21) − . Notice that u denotes the support function of M t given by u = h X , ν i and ϕ is a positive smooth function.The reason that we study the flow (1.1) is to explore the existence of the smooth solutions to the dualOrlicz-Minkowski problem introduced by Zhu-Xing-Ye [23], which is related to the following Monge-Amp`ere type equation u ϕ ( r ) r n + · det( u i j + u δ i j ) = f ( x ) on S n , (1.2)where r = p | Du | + u . In deed, let K be the set of all convex bodies in R n + which contain the origin intheir interiors, ϕ : (0 , + ∞ ) → (0 , + ∞ ) be a continuous function. Zhu-Xing-Ye [23] have introduced thedefinition of the dual Orlicz curvature measure e C ϕ ( K , · ), and posed the following dual Orlicz-Minkowskiproblem: Problem 1.1 (Dual Orlicz-Minkowski problem) . Under what conditions on ϕ and a nonzero finite Borelmeasure µ on S n , there exists a constant c > and a K ∈ K such that µ = c e C ϕ ( K , · ) ? When µ has a density f , this Minkowski problem is equivalent to solve the Monge-Amp`ere typeequation (1.2). When ϕ ( r ) = r q , this becomes the dual Minkowsi problem for the q -th dual curvatureconsidered by Huang-Lutwak-Yang-Zhang [16]. It is worth pointing out that they also proved the exis-tence of symmetric solutions for the case q ∈ (0 , n +
1) under some conditions. For q = n +
1, the dualMinkowski problem becomes the logarithmic Minkowski problem which studied in [8]. For q < ffi culty of our proof liesthe inhomogeneous term ϕ ( r ). To statement our theorem, we need the following assumption. Assumption 1.1. Φ : (0 , + ∞ ) → (0 , + ∞ ) is a continuous function such that Φ ( t ) = Z t ϕ ( s ) s dsexists for every t > . Theorem 1.2.
Assume that f ∈ C ∞ ( S n ) is a positive smooth function and ϕ : (0 , + ∞ ) → (0 , + ∞ ) is asmooth function. Let M ⊂ R n + be a strictly convex, closed hypersurface which contains the origin inits interior.(i) If max s > s ϕ ′ ( s ) ϕ − ( s ) < for any t ∈ (0 , + ∞ ) , then the normalised flow (1.1) has a unique smoothsolution, which exists for any time t ∈ [0 , ∞ ) . For each t ∈ [0 , ∞ ) , M t = X ( S n , t ) is a closed, smooth andstrictly convex hypersurface and the support function u ( x , t ) of M t = X ( S n , t ) converges smoothly, ast → ∞ , to the unique positive, smooth and strictly convex solution of the equation (1.2) with f replacedby λ f for some λ > .(ii) Under the assumption (1.1) , if f is in addition even function and the initial hypersurface M isorigin-symmetric, then the normalised flow (1.1) has a unique smooth solution, which exists for any timet ∈ [0 , ∞ ) . For each t ∈ [0 , ∞ ) , M t = X ( S n , t ) is a closed, smooth, strictly convex and origin-symmetrichypersurface and the support function u ( x , t ) of M t = X ( S n , t ) converges smoothly, as t → ∞ , to theunique positive, smooth, strictly convex and even solution of the equation (1.2) with f replaced by λ ffor some λ > . UAL ORLICZ MINKOWSKI PROBLEMS 3
Remark 1.2. If ϕ ( r ) = r q , the assumption max s > s ϕ ′ ( s ) ϕ − ( s ) < means q < , and the assumption (1.1) is equivalent to q ≥ , thus Theorem 1.2 recovers a parabolic proof in the smooth category for theexistence of solutions to the dual Minkowsi problem which given in [17] . The organization of this paper is as follows. In Sect. 2 we start with some preliminaries. In Sect. 3we obtain C and C estimates. The C estimates are given in Sect. 4. In Sect. 5 we prove Theorem 1.2.2. P reliminaries Setting and General facts.
For convenience, we first state our conventions on Riemann Curvature tensor and derivative notation.Let M be a smooth manifold and g be a Riemannian metric on M with Levi-Civita connection D . For a( s , r ) tensor field α on M , its covariant derivative D α is a ( s , r +
1) tensor field given by D α ( Y , .., Y s , X , ..., X r , X ) = D X α ( Y , .., Y s , X , ..., X r ) = X ( α ( Y , .., Y s , X , ..., X r )) − α ( D X Y , .., Y s , X , ..., X r ) − ... − α ( Y , .., Y s , X , ..., D X X r ) , the coordinate expression of which is denoted by D α = ( α l ··· l s k ··· k r ; k r + ) . We can continue to define the second covariant derivative of α as follows: D α ( Y , .., Y s , X , ..., X r , X , Y ) = ( D Y ( D α ))( Y , .., Y s , X , ..., X r , X ) , the coordinate expression of which is denoted by D α = ( α l ··· l s k ··· k r ; k r + k r + ) . Similarly, we can also define the higher order covariant derivative of α : D α = D ( D α ) , ... and so on. For simplicity, the coordinate expression of the covariant di ff erentiation will usually bedenoted by indices without semicolons, e.g. u i , u i j or u i jk for a function u : M → R .Our convention for the Riemannian curvature (3,1)-tensor Rm is defined by Rm ( X , Y ) Z = − D X D Y Z + D Y D X Z + D [ X , Y ] Z . Pick a local coordinate chart { x i } ni = of M . The component of the (3,1)-tensor Rm is defined by Rm (cid:18) ∂∂ x i , ∂∂ x j (cid:19) ∂∂ x k (cid:17) R li jk ∂∂ x l and R i jkl = g lm R mi jk . Then, we have the standard commutation formulas (Ricci identities): α l ··· l s k ··· k r ; ji − α l ··· l s k ··· k r ; i j = r X a = R mi jk l α l ··· l s k ··· k a − mk a + ··· k r − s X b = R l b i jm α l ··· l b − ml b + ··· l r k ··· k r . (2.1)We list some facts which will be used frequently. For the standard sphere S n with the sectional curvature1, R i jkl = δ ik δ jl − δ il δ jk . A special case of Ricci identity for a function u : M → R will be usually used frequently: u k ji − u ki j = R mi jk u m . LI CHEN, QIANG TU, DI WU, NI XIANG
In particular, for a function u : S n → R , u k ji − u ki j = δ ik u j − δ jk u i . (2.2)Let ( M , g ) be an immersed hypersurface in R n + and ν be a given unit outward normal. The secondfundamental form h i j of the hypersurface M with respect to ν is defined by h i j = − * ∂ X ∂ x i ∂ x j , ν + R n + . Basic properties of convex hypersurfaces.
We first recall some basic properties of convex hypersurfaces. Let M be a smooth, closed, uniformlyconvex hypersurface in R n + . Assume that M is parametrized by the inverse Gauss map X : S n → M . The support function u : S n → R of M is defined by u ( x ) = sup {h x , y i : y ∈ M} . The supremum is attained at a point y such that x is the outer normal of M at X . It is easy to check that X = u ( x ) x + Du ( x ) , where D is the covariant derivative with respect to the standard metric σ i j of the sphere S n . Hence r = | X | = p u + | Du | . (2.3)Thus, u = r p r + | Dr | . (2.4)The second fundamental form of M is given by, see e.g. [2, 20], h i j = u i j + σ i j , (2.5)where u i j = D i D j u denotes the second order covariant derivative of u with respect to the spherical metric σ i j . By Weingarten’s formula, σ i j = h ∂ν∂ x i , ∂ν∂ x j i = h ik g kl h jl , (2.6)where g i j is the metric of M and g i j is its inverse. It follows from (2.5) and (2.6) that the principal radiiof curvature of M , under a smooth local orthonormal frame on S n , are the eigenvalues of the matrix b i j = u i j + u δ i j . In particular, the Gauss curvature is given by K = u i j + u δ i j ) . UAL ORLICZ MINKOWSKI PROBLEMS 5
Geometric flow and its associated functional.
For reader’ convenience, the associated Mong-Amp`ere equation (1.2) is restated here, u ϕ ( r ) r n + · det( u i j + u δ i j ) = f ( x ) on S n . Recall the normalised anisotropic Gauss curvature flow (1.1) ∂ X ∂ t = − θ ( t ) f ( ν ) r n + ϕ ( r ) K ν + X , X ( · , = X , where θ ( t ) = Z S n ϕ ( r ( ξ, t )) d ξ (cid:20) Z S n f ( x ) dx (cid:21) − . By the definition of support function, we know u ( x , t ) = h x , X ( x , t ) i . Hence,(2.7) ∂ u ∂ t ( x , t ) = − θ ( t ) f ( x ) r n + ϕ ( r ) K + u ( x , t ) , u ( · , = u . The normalised flow (1.1) can be also described by the following scalar equation for r ( · , t )(2.8) ∂ r ∂ t ( ξ, t ) = − θ ( t ) f ( x ) r n + ϕ ( r ) u K + r ( ξ, t ) , r ( · , = r , in view of 1 r ( ξ, t ) ∂ r ( ξ, t ) ∂ t = u ( x , t ) ∂ u ( x , t ) ∂ t , see Section 3 in [10] for the proof.For a convex body Ω ⊂ R n + , we define V ϕ ( Ω ) = Z S n d ξ Z r ( ξ, t )0 ϕ ( s ) s ds . When ϕ ( s ) = s q , V ϕ ( Ω ) be the q -volume of the convex body Ω ⊂ R n + , see [9, 10]. We show below that V ϕ ( Ω t ) is unchanged under the flow (1.1), where Ω t is a compact convex body in R n + with the boundary M t . Lemma 2.1.
Let X ( · , t ) be a strictly convex solution to the flow (1.1) , then we obtainV ϕ ( Ω t ) = V ϕ ( Ω ) . Proof. ddt V ϕ ( Ω t ) = Z S n ϕ ( r ) r ∂ r ∂ t d ξ = Z S n ϕ ( r ) r (cid:18) − θ ( t ) f ( x ) r n + ϕ ( r ) u K + r ( ξ, t )) (cid:19) d ξ = − θ ( t ) Z S n f ( x ) r n + u Kd ξ + Z S n ϕ ( r ) d ξ = , where we use dxd ξ = r n + Ku , LI CHEN, QIANG TU, DI WU, NI XIANG see e.g. [10, 16]. (cid:3)
Next, we define the functional J ϕ ( X ( · , t )) = Z S n log u ( x , t ) f ( x ) dx . The following lemma shows that the functional J ϕ is non-increasing along the flow (1.1). Lemma 2.2.
Let X ( · , t ) be a strictly convex solution to the flow (1.1) . For any ϕ ≥ , the functional isnon-increasing along the flow (1.1) . In particular,ddt J ϕ ( X ( · , t )) ≤ . and the equality holds if and only if X t satisfies the elliptic equation (1.2) with f replaced by θ ( t ) f .Proof. ddt J ϕ ( X ( · , t )) = Z S n u ∂ u ( x , t ) ∂ t f ( x ) dx = Z S n u (cid:18) − θ ( t ) f ( x ) r n + ϕ ( r ) K + u ( x , t ) (cid:19) f ( x ) dx = (cid:20) Z S n f ( x ) dx (cid:21) − (cid:26) − Z S n u ϕ ( r ) r n + K dx Z S n r n + Ku ϕ ( r ) f dx + Z S n f dx Z S n f dx (cid:27) = (cid:20) Z S n f ( x ) dx (cid:21) − (cid:26) − Z S n u ϕ ( r ) f r n + K d σ Z S n r n + Ku ϕ ( r ) f d σ + Z S n d σ Z S n d σ (cid:27) ≤ Z S n d σ Z S n d σ ≤ Z S n u ϕ ( r ) f r n + K d σ Z S n r n + Ku ϕ ( r ) f d σ, which is implies by H ¨ o lder inequality, where d σ = f ( x ) dx . Clearly, the equality holds if and only if f ( x ) r n + Ku ϕ ( r ) = c ( t ) . In this case, clearly, we have θ ( t ) = c ( t ). Thus, X ( · , t ) satisfies the elliptic equation (1.2) with f replacedby θ ( t ) f . (cid:3) Before closing this section, we prove the following basic properties for any given Ω ∈ K , whilesmoothness of ∂ Ω is not required. First, we introduce the following Lemma for convex bodies, seeLemma 2.6 in [10] for the details. Lemma 2.3.
Let Ω ∈ K . Let u and r be the support function and radial function of Ω , and x max and ξ min be two points such that u ( x max ) = max S n u and r ( ξ min ) = min S n r. Then max S n u = max S n r and min S n u = min S n r , u ( x ) ≥ x · x max u ( x max ) , ∀ x ∈ S n , r ( ξ ) ξ · ξ min ≥ r ( ξ min ) , ∀ ξ ∈ S n . Let K n + = { K | K is convex body in R n + } . Then, we have the following theorem (see also [18]). UAL ORLICZ MINKOWSKI PROBLEMS 7
Theorem 2.4.
If K i ∈ K n + and there exists a constant R > such that K i ⊂ B R , then there exists asubsequence K i j and K ∈ K n + such thatK i j → K in the Hausdor ff metric . To statement the following theorem, we first recall the definition of the radial function of a convexbody. (see also [18]).
Definition 2.1.
Let K ∈ K n + , ∈ K, a radial function r K : R n + \{ } → R is defined asr K ( x ) = max { r ≥ | rx ∈ K } . Now, the convergence of convex bodies imply the convergence of the corresponding radial functions.
Theorem 2.5.
Let K , K i ∈ K n + , ∈ intK and K i → K , then r K i ⇒ r K . For the proof of the theorem above, see [18].3. C , C - estimates In this section, we will derive the C , C -estimates of the flow (1.1). The key is the lower bound of u .The di ffi culty of the proof lies the inhomogeneous term ϕ ( r ).3.1. The upper bound of u and gradient estimate. It is easy to obtain the upper bound of u andgradient estimate if we notice that the functional J ϕ is non-increasing along the flow (1.1), see Lemma2.2. Lemma 3.1.
Let X ( · , t ) be a strictly convex solution to the flow (1.1) , then we haveu ( · , t ) ≤ C , ∀ t ∈ [0 , T ) . (3.1) and | Du | ( · , t ) ≤ C , ∀ t ∈ [0 , T ) . (3.2) Proof.
Assume that x t is a point at where u ( · , t ) attains its spatial maximum, we know from Lemma 2.2 C ≥ Z S n log u ( x , t ) f ( x ) dx ≥ Z { x ∈ S n : x · x t > } log[ x · x t u ( x t , t )] f ( x ) dx , which implies C ≥ max S n u ( · , t ) . This yields the inequality (3.1). Sincemax S n | Du | ( · , t ) ≤ max S n u ( · , t ) , we obtain (3.2). (cid:3) The lower bound of u . We get the lower bound of u by the following gradient estimate for Case(i) in Theorem 1.2 and the fact that f and u are even functions for Case (ii) in Theorem 1.2. Lemma 3.2.
Let X ( · , t ) be a strictly convex solution to the flow (1.1) , if max s > s ϕ ′ ( s ) ϕ − ( s ) < , (3.3) then max S n | Du | u ( · , t ) ≤ C , ∀ t ∈ [0 , T ) . (3.4) LI CHEN, QIANG TU, DI WU, NI XIANG
Proof.
Let z = log u , it is straightforward to see ∂ z ∂ t = − θ ( t ) f ( x ) (1 + | Dz | ) n + ϕ ( e z p + | Dz | ) 1det( z i j + z i z j + δ i j ) + = Q ( D z , Dz , z ) + . Set ψ = | Dz | . By di ff erentiating the ψ ,we have ∂ψ∂ t = ( ∂∂ t z m ) z m = (˙ z ) m z m = Q m z m . Then, ∂ψ∂ t = Q i j z i jm z m + Q k z km z m + ( − e z p + | Dz | ϕ ′ ϕ − Q | Dz | + h D log f , Dz i Q ) . where Q i j = ∂ Q ∂ w i j = − Qw i j , Q k = ∂ Q ∂ z k . Interchanging the covariant derivatives, we have ψ i j = ( z mi z m ) j = z mi j z m + z mi z mj = z im j z m + z mi z mj = z i jm z m + σ i j | Dz | − z i z j + z mi z mj in view of (2.2). Thus, we have(3.5) ∂ψ∂ t = Q i j ψ i j + Q k ψ k − Q i j ( δ i j | Dz | − z i z j ) − Q i j z mi z mj + ( − e z p + | Dz | ϕ ′ ϕ − | Dz | + h D log f , Dz i ) Q ≤ Q i j ψ i j + Q k ψ k − Q i j ( δ i j | Dz | − z i z j ) − Q i j z mi z mj + ( − e z p + | Dz | ϕ ′ ϕ − | Dz | − C ) Q | Dz | . Since the matrix Q i j and δ i j | D ϕ | − ϕ i ϕ j are positive definite, the third and forth terms in the right of(3.5) are non-positive. And noticing that the fifth term in the right of (3.5) is nonpositive if (3.3) holdstrue and | Dz | ≥ − C max s > s ϕ ′ ( s ) ϕ − ( s ) . So we got the equation about ψ as follows: ∂ψ∂ t ≤ Q i j ψ i j + Q k ψ k on S n × (0 , ∞ ) ,ψ ( · , = | Dz ( · , | on S n . Using the maximum principle, we get the gradient estimates of z . (cid:3) Lemma 3.3.
Let X ( · , t ) be a strictly convex solution to the flow (1.1) , then we have C ≤ u ( x , t ) ≤ C , ∀ ( x , t ) ∈ S n × [0 , T ) . (3.6) if either (i) (3.3) holds true; or (ii) the assumption 1.1 holds true, f and u are even functions. UAL ORLICZ MINKOWSKI PROBLEMS 9
Proof.
We only need prove the first inequality in (3.6) by noticing Lemma 3.1.Case (i): If (3.3) holds true, we have by virtue of (3.4)max S n log u ( · , t ) − min S n log u ( · , t ) ≤ C max S n | Du | u ( · , t ) ≤ C , which implies the positive lower bound of u together with (3.1).Case (ii): f and u are even. We have(3.7) Z S n d ξ Z r ( ξ, ϕ ( s ) s ds = Z S n d ξ Z r ( ξ, t )0 ϕ ( s ) s ds by Lemma 2.1. Here we use the idea in [9] to complete our proof by contradiction. Assume r ( ξ, t ) isnot uniformly bounded away from 0 which means there exists inf x ∈ S n r ( ξ, t i ) → i → ∞ , where t i ∈ [0 , T ). Since f and u are even, r ( ξ, t ) is even. Thus, Ω t is a origin-symmetric body, where Ω t is theconvex body containing the origin and ∂ Ω t = M t . Thus, using Theorem 2.4, we have Ω t i (after choosinga subsequence) converges to a origin-symmetric convex body Ω . Then, we have by Theorem 2.5inf ξ ∈ S n r Ω ( ξ ) = . So, there exists ξ ∈ S n such that r Ω ( ξ ) = r Ω ( − ξ ) =
0, which implies Ω contained in alower-dimensional subspace. This means that r ( ξ, t i ) → i → ∞ almost everywhere with respect to the spherical Lebesgue measure. Combined with boundedconvergence theorem, we conclude Z S n d ξ Z r ( ξ, ϕ ( s ) s ds = Z S n d ξ Z r ( ξ, t i )0 ϕ ( s ) s ds → i → ∞ , which is a contraction to (3.7). So, we complete our proof. (cid:3) The C and C estimates of u imply the corresponding C and C estimates of r by using (2.4) andLemma 2.3. Corollary 3.4.
Under the assumptions in Theorem 1.2, if X ( · , t ) is a strictly convex solution to the flow (1.1) , then we have C ≤ r ( ξ, t ) ≤ C , ∀ ( ξ, t ) ∈ S n × [0 , T ) , (3.8) | Dr | ( ξ, t ) ≤ C , ∀ ( ξ, t ) ∈ S n × [0 , T ) , (3.9) and C ≤ θ ( t ) ≤ C , ∀ t ∈ [0 , T ) . (3.10) 4. C - estimates In this section we establish uniformly positive and lower bounds for the principle curvatures for thenormalised flow (1.1). We first use the technique that was first introduced by Tso [19] to derive the upperbound of the Gauss curvature along the flow (1.1), see also the proof of Lemma 4.1 in [17] and Lemma5.1 in [9].
Lemma 4.1.
Let X ( · , t ) be a strictly convex solution to the flow (1.1) which encloses the origin fort ∈ [0 , T ) . Then, there exists a positive constant C depending only ϕ , max S n × [0 , T ) u and min S n × [0 , T ) u,such that max S n K ( · , t ) ≤ C , ∀ t ∈ [0 , T ) . Proof.
We apply the maximum principle to the following auxiliary function defined on the unit sphere S n W ( x , t ) = θ ( t ) − u t + uu − ε = f ( x ) ϕ ( r ) r n + Ku − ε , where ε =
12 min ( x , t ) ∈ S n × [0 , T ) u ( x , t ) > . At the maximum x of W for any fixed t ∈ [0 , T ), we have at ( x , t )0 = θ ( t ) W i = − u ti + u i u − ε + u t − u ( u − ε ) u i , (4.1)and 0 ≥ θ ( t ) D i j W = − u ti j + u i j u − ε + ( u t − u ) u i j ( u − ε ) , (4.2)where (4.1) was used in deriving the second equality above. The inequality (4.2) should be understoodin sense of positive-semidefinite matrix. Hence, u ti j + u t δ i j ≥ θ ( t )( − b i j + ε δ i j ) W + b i j . Thus, K t = − Kb i j ( u ti j + u t δ i j ) ≤ − nK − θ ( t ) KW ( − n + ε H ) , where H denotes the mean curvature of X ( · , t ). Noticing that H ≥ nK n , we obtain K t ≤ CW (1 + W ) − CW + n . Using the equation (2.7) and the inequality above, we have W t = (cid:20) f ( x ) ϕ ( r ) r n + u − ε (cid:21) t K + (cid:20) f ( x ) ϕ ( r ) r n + u − ε (cid:21) K t ≤ CW + CW − CW + n , in view of u t ≈ CW + C , r t = uu t + u k u kt r ≈ CW + C . Without loss of generality we assume that K ≈ W ≫
1, which implies that W t ≤ . Therefore, we arrive at W ≤ C for some constant C > C -norm of r and ε . Thus,the priori bound follows consequently. (cid:3) Now, we show the principle curvatures of X ( · , t ) are bounded from below along the flow (1.1). Theproof is similar to Lemma 4.2 in [17] and Lemma 5.1 in [9]. Lemma 4.2.
Let X ( · , t ) be a strictly convex solution to the flow (1.1) which encloses the origin fort ∈ [0 , T ) . Then, there exists a positive constant C depending only ϕ , q, max S n × [0 , T ) u and min S n × [0 , T ) u,such that the principle curvatures of X ( · , t ) are bounded from below κ i ( x , t ) ≥ C , ∀ ( x , t ) ∈ S n × [0 , T ) , and i = , ..., n . (4.3) UAL ORLICZ MINKOWSKI PROBLEMS 11
Proof.
We consider the auxiliary function e Λ ( x , t ) = log λ max ( { b i j } ) − A log u + B | Du | , where A and B are positive constants which will be chosen later, and λ max ( { b i j } ) denotes the maximaleigenvalue of { b i j } . For convenience, we write { b i j } for { b i j } − .For any fixed t ∈ [0 , T ), we assume the maximum e Λ is achieved at some point x ∈ S n . By rotation,we may assume { b i j ( x , t ) } is diagonal and λ max ( { b i j } )( x , t ) = b ( x , t ). Thus, it is su ffi cient to prove b ( x , t ) ≤ C .Then, we define a new auxiliary function Λ ( x , t ) = log b − A log u + B | Du | , which attains the local maximum at x for fixed time t . Thus, we have at x = D i Λ = b b i − A u i u + B X k u k u ki (4.4)and 0 ≥ D i D j Λ = b b i j − ( b ) b i b j − A (cid:18) u i j u − u i u j u (cid:19) + B X k (cid:18) u k j u ki + u k u ki j (cid:19) . (4.5)We can rewrite the equation (2.7) aslog( u − u t ) = − log det( b ) + α ( x , t ) , (4.6)where α ( x , t ) = log (cid:18) θ ( t ) f ( x ) r n + ϕ ( r ) (cid:19) . Di ff erentiating (4.6) gives u k − u kt u − u t = − b i j b i j ; k + D k α (4.7)and u − u t u − u t = ( u − u t ) ( u − u t ) − b ii b ii ;11 + b ii b j j ( b i j ;1 ) + D D α. (4.8)Recalling the Ricci identity (2.1) b ii ;11 = b ii − b + b ii , which is taken into (4.8) implies u − u t u − u t = ( u − u t ) ( u − u t ) − b ii b ii + X i b ii b − n + b ii b j j ( b i j ;1 ) + D D α. (4.9)So, we have ∂ t Λ u − u t = b (cid:18) u t − u u − u t + u + u − u + u t u − u t (cid:19) − A u u t − u + uu − u t + B u k u kt u − u t (4.10) = b (cid:20) − ( u − u t ) ( u − u t ) + b ii b ii − X i b ii b − b ii b j j ( b i j ;1 ) − D D α (cid:21) + − Au − u t + Au + B P k u k u kt u − u t + ( n − b . We know from (4.5) and (4.7)0 ≥ b [ b ii b ii − b ii b ( b i ) ] − A nu + A X i b ii + Ab ii u i u i u
22 LI CHEN, QIANG TU, DI WU, NI XIANG + B (cid:20) b ii ( b ii − u ) + X k u k ( D k α − u k − u kt u − u t ) − b ii u i u i (cid:21) ≥ b [ b ii b ii − b ii b j j ( b i j ;1 ) ] − A nu + A X i b ii + Ab ii u i u i u + B (cid:20) X i b ii ( b ii − ub ii ) + X k u k ( D k α − u k − u kt u − u t ) − b ii u i u i (cid:21) ≥ b [ b ii b ii − b ii b j j ( b i j ;1 ) ] − A nu + A X i b ii + Ab ii u i u i u + B (cid:20) X i b ii − nu + X k u k ( D k α − u k − u kt u − u t ) − b ii u i u i (cid:21) . Thus, plugging the inequality above into (4.10) gives ∂ t Λ u − u t ≤ − b D D α − B X k u k D k α + − A + B | Du | u − u t (4.11) + ( n + Au + ( n − b + (2 B | Du | − A − X i b ii − Ab ii u i u i u − B X i b ii + nBu . Now, we need estimate the first two terms in the inequality above. Clearly, a direct calculation results in r i = uu i + P k u k u ki r = u i b ii r and r i j = uu i j + u i u j + P k u k u ki j + P k u k j u ki r − u i u i b ii b j j r . Hence, we obtain by Lemma 3.1, Lemma 3.3 and Corollary 3.4 − b D D α − B X k u k D k α = − b (cid:20) f f − f f − ( n + r r + ( ϕ ′ ) r ϕ − ϕ ′′ r ϕ (cid:21) − b (cid:20) ( n +
1) 1 r − ϕ ′ ϕ (cid:21) r − B X k u k (cid:18) f k f + [( n +
1) 1 r − ϕ ′ ϕ ] r k (cid:19) ≤ Cb (1 + b ) + CB − (cid:20) ( n +
1) 1 r − ϕ ′ ϕ (cid:21) ( b r + Bu k r k ) ≤ Cb (1 + b + b ) + CB − (cid:20) ( n +
1) 1 r − ϕ ′ ϕ (cid:21)(cid:18) b u k u k r + B u k u k u kk r (cid:19) . Then, using (4.4), we have − b D D α − B X k u k D k α ≤ Cb (1 + b + b ) + CB − (cid:20) ( n +
1) 1 r − ϕ ′ ϕ (cid:21) u k r (cid:18) A u k u − b u δ k (cid:19) ≤ Cb (1 + b + b ) + CB + CA . UAL ORLICZ MINKOWSKI PROBLEMS 13
Thus, using the inequality above, we conclude from (4.11) ∂ t Λ u − u t ≤ C ( b + + b ) + CB + CA + ( n + Au + ( n − b + (2 B | Du | − A − X i b ii − Ab ii u i u i u − B X i b ii + nBu < , provided b ≫ A ≫ B . So we complete the proof. (cid:3)
5. T he convergence of the normalised flow
With the help of a prior estimates in the section above, we show the long-time existence and asymp-totic behaviour of the normalised flow (1.1) which complete Theorem 1.2.
Proof.
Since the equation (2.7) is parabolic, we have the short time existence. Let T be the maximal timesuch that u ( · , t ) is a positive, smooth and strictly convex solution to (2.7) for all t ∈ [0 , T ). Lemmas 3.1,3.2, 4.1 and Corollary 3.4 enable us to apply Lemma 4.2 to the equation (2.7) and thus we can deduce auniformly lower estimate for the biggest eigenvalue of { ( u i j + u δ i j )( x , t ) } . This together with Lemma 4.2implies C − I ≤ ( u i j + u δ i j )( x , t ) ≤ CI , ∀ ( x , t ) ∈ S n × [0 , T ) , where C > n , α, f and u . This shows that the equation (2.7) is uniformly parabolic.Using Evans-Krylov estimates and Schauder estimates, we obtain | u | C l , mx , t ( S n × [0 , T )) ≤ C l , m for some C l , m independent of T . Hence T = ∞ . The uniqueness of the smooth solution u ( · , t ) follows bythe parabolic comparison principle.By the monotonicity of J ϕ (See Lemma 2.2), and noticing that |J ϕ ( X ( · , t )) | ≤ C , ∀ t ∈ [0 , ∞ ) , we conclude that Z ∞ | ddt J ϕ ( X ( · , t )) | ≤ C . Hence, there is a sequence t i → ∞ such that ddt J ϕ ( X ( · , t i )) → . In view of Lemma 2.2, we see that u ( · , t i ) converges smoothly to a positive, smooth and strictly convex u ∞ solving (1.2) with f replaced by λ f with λ = lim t i →∞ θ ( t i ). (cid:3) R eferences [1] B. Andrews, Evolving convex curves , Calc. Vara. Part. Di ff . Equa., (1998), 315-371.[2] B. Andrews, Motion of hypersurfaces by Gauss curvature , Pacific J. Math., (2000), 1-34.[3] B. Andrews,
Gauss curvature flow: the fate of the rolling stones , Invent. Math., (1999), 151-161.[4] B. Andrews,
Classification of limiting shapes for isotropic curve flows , J. Amer. Math. Soc., (2003), 443-459.[5] B. Andrews and X. Chen, Surface moving by powers of Gauss curvature , Pure. Appl. Math. Q., (2012),825-834. [6] Ben Andrews, Contraction of convex hypersurfaces by their a ffi ne normal, J. Di ff erential Geom. 43 (1996),no. 2, 207-230.[7] Ben Andrews, Pengfei Guan, and Lei Ni, Flow by powers of the Gauss curvature, Adv. Math. 299 (2016),174-201.[8] k. B ¨ o r ¨ o czky, E. Lutwak, D. Yang, G. Zhang, The logarithmic Minkowski problem , J. Amer. Math. Soc., (2013), 831-852.[9] C. Chen, Y. Huang and Y. Zhao, Smooth solutions to the L p dual Minkowski problem , Math. Ann., (2019),953-976.[10] H. Chen and Q. Li, The L p dual Minkowski problems and related parabolic flows , preprint.[11] S.Brendle, K. Choi, and P. Daskalopoulos, Asymptotic behavior of flows by powers of the Gaussian curvature ,Acta. Math., (2017), 1-16.[12] B. Chow,
Deforming convex hypersurfaces by the n-th root of the Gaussian curvature , J. Di ff . Geom., (1985), 117-138.[13] K. Chou and X. Wang, A logarithmic Gauss curvature flow and the Minkowski problem , Ann. Inst. H.Poincare Anal. Non Lineaire, (2000), 733-751.[14] W. Firey, Shapes of worn stones , Mathematika, (1974), 1-11.[15] Pengfei Guan and Lei Ni, Entropy and a convergence theorem for Gauss curvature ?ow in high dimension, J.Eur. Math. Soc. (JEMS) 19 (2017), no. 12, 3735C3761.[16] Y. Huang, E. Lutwak, D. Yang and G. Zhang, Geometric measures in the dual Brunn-Minkowski theory andtheir associated Minkowski problems , Acta Math., (2016), 325-388.[17] Q. Li, W. Sheng and X. Wang,
Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems ,J. Eur. Math. Soc., arXiv:1712.07774, 2017.[18] R. Schneider,
Convex bodies: the Brunn-Minkowski theory , Second edition, No. 151. Cambr. Univ. Press,2013.[19] K. Tso,
Deforming a hypersurface by its Gauss-Kronecker curvature , Comm. Pure Appl. Math., (1985),867-882.[20] J. Urbas, An expansion of convex hypersurfaces , J. Di ff . Geom., (1991), 91-125.[21] X. Wang, Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature , Trans. Amer. Math.Soc., (1996), 4501-4524.[22] Y. Zhao,
The Dual Minkowski Problem for Negative Indices , Calc. Var. PDEs, (2017), Art. 18, 16 pp.[23] B. Zhu, S. Xing and D. Ye, The dual Orlicz-Minkowski problem , J. Geom. Anal., (2015), 810-829. H ubei K ey L aboratory of A pplied M athematics , F aculty of M athematics and S tatistics , H ubei U niversity , W uhan hina E-mail address ::