Orlicz-Minkowski flows

We study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if exist, solve the regular Orlicz-Minkowski problems. As an application, we obtain old and new results for the regular even Orlicz-Minkowski problems; the corresponding $L_p$ version is the even $L_p$-Minkowski problem for $p>-n-1$. Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz-Minkowski problems; the $L_p$ versions are the even $L_p$-Minkowski problem for $p>0$ and the $L_p$-Minkowski problem for $p>1$. In the final section, we use a curvature flow with no global term to solve a class of $L_p$-Christoffel-Minkowski type problems.


Introduction
A convex body in Euclidean space is a compact convex set with nonempty interior. Write K for set of convex bodies, and K o for the set of convex bodies containing the origin o. The support function of a convex body K is defined by For any u on the boundary of K, let ν K (u) be the set of all unit exterior normal vectors at u. The surface area measure of K, S K , is a Borel measure on the unit sphere defined by S K (ω) = H n (ν −1 K (ω)) for all Borel sets ω ⊂ S n . Here H n denotes the n-dimensional Hausdorff measure. When the boundary of K, ∂K, is a C 2 smooth, strictly convex hypersurface, then dS K = σ n dθ, where σ n is the product of the principal radii of curvature and dθ is the standard spherical Lebesgue measure. In this case, h K , as a function on the unit sphere, is given by . The Minkowski problem is one of the cornerstones of the classical Brunn-Minkowski theory. It asks what are the necessary and sufficient conditions on a Borel measure µ on S n in order for it to be the surface area measure of a convex body. The complete solution to this problem was found by Minkowski, Aleksandrov and Fenchel-Jessen (see [Sch13]): A Borel measure µ whose support is not contained in a closed hemisphere is the surface area measure of a convex body if and only if S n udµ(u) = o.
Moreover, the solution is unique up to translations.
Let ϕ : (0, ∞) → (0, ∞) be a continuous function. The general Orlicz-Minkowski problem asks what are the necessary and sufficient conditions on a Borel measure µ on S n , such that there exists a convex body K ∈ K o so that ϕ(h K )dS K = γdµ for some constant γ > 0.
A natural generalization of L p spaces are Orlicz spaces, motivating the Orlicz linear combination of convex bodies and leading to the Orlicz-Minkowski problem. We keep the discussion brief here and the reader may consult [BBC19, HLYZ10, GHW14, GHW + 19, GHXY20] as well as [XJL14,LYZ10,Sch13,ZX14] for the origin of the Orlicz-Brunn-Minkowski theory and related concepts.
The regular Orlicz-Minkowski problem asks what are the necessary and sufficient conditions on the smooth function f : S n → (0, ∞) such that there exists a convex hypersurface with support function h (as a function on the unit sphere) satisfying f ϕ(h)σ n (h ij + δ ij h) = γ for some constant γ > 0. (1.2) The subscripts of h denote covariant derivatives with respect to a local orthonormal frame field on S n . Our approach to solving this problem is via the flow method in the regular case (1.2) and by parabolic approximation in the general case (1.1).

Curvature flows. Let
x 0 : M n → R n+1 be a smooth parametrization of a closed, strictly convex hypersurface M 0 with the origin of R n+1 in its interior. One of the flows that we are interested in is the family of hypersurfaces {M t } given by the smooth map x : satisfying the initial value problem where K(·, t) and ν(·, t) are respectively the Gauss curvature and the outer unit normal vector of M t := x(M n , t), f : S n → (0, ∞) and ϕ : (0, ∞) → (0, ∞) are smooth functions and To explain our interest in studying this flow, we need some definitions and notation.
By a direct calculation, along the flow (2.1) the support functions of the M t (as long as they are strictly convex) satisfy Stationary solutions of (2.2), whenever they exist, are exactly the solutions of the regular Orlicz-Minkowski problem and thus the limits of the flow hypersurfaces are solutions of (1.2).
For some choices of f and ϕ(s) = s 1−p , the flow (2.1) becomes homogeneous and was considered in [And98, BIS18, CT97, Ger14, Iva19a, Iva16, Iva15, Iva14, IS13, Li10, Sch16, Sch06, Sta12]. However, when it comes to non-homogeneous flows, the literature on geometric flows is not very rich and there are few works in this direction; e.g., [BS18, CT98, CT97, CW00, Krö17, LZ19, LL20]. Since the Orlicz-Minkowski problem admits solutions in the non-homogeneous case, it is desirable to remove the homogeneity assumption in the flow. The conditions we place on f, ϕ are similar to those used previously in the literature on the Orlicz-Minkowski problem mentioned in the introduction while some are new (Remark 2.2 and Remark 2.4 below).
(2) Since φ as defined in (3-a) is increasing, we havê Therefore, the integral conditions in (3-b) and (3-d) compared to the condition (3-a) are not more restrictive. (3) The cases (3-b) and (3-d) yield new existence results for the regular even Orlicz-Minkowski problem, while the existence results corresponding to (3-a) and (3-c) can be deduced from [BBC19,HLYZ10]. (4) Due to the presence of ϕ (possibly non-homogenous) and ζ in the speed of the flow (2.1), we could not employ the method of [And97] to promote subconvergence to full convergence.
We slightly modify the flow (2.1) to treat a class of regular Orlicz-Minkowski problems without the assumption that f is even, i.e., we no longer assume In order to serve this purpose, we consider the flow where again x 0 : M n → R n+1 is a smooth parametrization of a closed, strictly convex hypersurface M 0 with the origin of R n+1 in its interior. Here compared to (2.1), we dropped the ζ-factor. In the appendix, we will use another curvature flow without a global term (5.1) to treat a class of L p -Minkowski-Christoffel type problems. See also [BIS20,Ger96,Ger03,Ger06,Ger08] for flows without global terms.
. Then there exists a smooth, strictly convex solution M t of (2.3) and it converges in C ∞ to a smooth, strictly convex solution of (1.2) with positive support function and constant γ = 1.
(1) In the case special case ϕ(s) = s 1−p , Theorem 2.3 finds the solutions of the regular L p -Minkowski problems for p > n + 1.
(2) In [CW00, p. 42], using a logarithmic curvature flow, an existence result was obtained under the assumption (2) and that ϕ is non-increasing. See also [LL20], where a similar result was recently obtained.

General measures.
Definition 2.5. A Borel measure µ on S n is said to be even if it assumes the same values on antipodal Borel sets. We say µ is invariant under a subgroup G of the orthogonal group O(n + 1) if µ(gω) = µ(ω) for all Borel sets ω ⊆ S n . Let µ be a finite even Borel measure on S n whose support is not contained on a great subsphere. Then there exists an o-symmetric convex body such that This theorem was first proved in [HLYZ10, Thm. 2] and contains the general even L p -Minkowski problem for p > 0. The method of HLYZ is a variational argument that finds a minimizing body of a suitable functional in a certain class of origin-symmetric (o-symmetric) convex bodies. We treated the regular version of this theorem by using the curvature flow in Theorem 2.1. The general case will be treated by a simple parabolic approximation, cf., Section 4.2.
In the next theorem, the assumption that the measure µ is even is dropped. To prove this theorem, we use the flow (2.1) and a version of Chou-Wang's approximation argument [CW00] adapted to the parabolic setting [CL19] for treating the general L p -Minkowski problem.
Let µ be a finite Borel measure on S n whose support is not contained in a closed hemisphere. Then there exist K ∈ K o and a constant γ > 0 such that Moreover, if µ is invariant under a closed group G ⊂ O(n + 1), then K can be chosen to be invariant under G.
This theorem first appeared in [HH12] and for the special case ϕ(s) = s 1−p , it includes the L p -Minkowski problem for p > 1. In [HH12], the problem is first solved for discrete measures and then by approximating the general measure by discrete measures; cf., Section 4.2 for the sketch and the full details. We solve it first in the regular case and then by approximating the general measures by the regular case. See also Wu-Xi-Leng [WXL19, WXL18] for the discrete case of the Orlicz-Minkowski problem, and [JL19] for an existence result of solutions to general Orlicz-Minkowski problem which contains the L p -Minkowski problem for 0 < p < 1. Note that the role of the constant γ is essential in general; see [YY15] for a non-existence result when γ is dropped.

Flow hypersurfaces
3.1. Regularity estimates. In the sequel,ḡ and∇ denote respectively the standard round metric and the Levi-Civita connection of S n . The principal radii of curvature are the eigenvalues of the matrix with respect toḡ. For convenience, we put Lemma 3.1. The following evolution equations hold along the flows.
Proof. We have ∂ t h = Θσ n − ηh and hence the first equation follows from the n-homogeneity and (3.1). For the second equation, in an orthonormal frame that diagonalizes r ij , Tracing with respect to σ ab n gives σ ab n∇ a∇b

Now the second evolution equation follows from
Finally there holds Lemma 3.2. Along the flow (2.1) we have Here V denotes the enclosed volume of M t . In both cases, the monotonicity is strict unless the solution is stationary.
For the flow (2.3), note that The equality holds precisely when h solves (1.2) with γ = 1.
We write w − and w + respectively for the minimum width and the maximum width of a closed, convex hypersurface (or a convex body) with support function h. They are defined as Moreover, η(t) is uniformly bounded above, and below away from zero.
Proof. Suppose R := max h Mt is attained at the north pole e n+1 . Theorem 2.1, case (3-a): Due to convexity, Since φ ≥ 0 is non-decreasing and E(t) constant in time, we obtain Due to lim s→∞ φ(s) = ∞, we see that h(·, t) is uniformly bounded above. Let V (M t ) denote the volume of the enclosed region by M t . To prove the uniform lower bound of h, note that Since M t is o-symmetric, w + = 2 max h and w − = 2 min h. The lower bound of h now follows since V (M t ) is non-decreasing along the flow. The lower and upper bounds on h imply bounds on η.
Theorem 2.1, case (3-b): Suppose R > 2. Due to convexity, and E(t) constant in time. Hence, Since lim s→∞ φ(s) = ∞, R remains uniformly bounded above. This in turn implies the lower and upper bounds on support functions and η. Theorem 2.1, case (3-d): Suppose R > 1. Since φ ≥ 0 is nonincreasing and E(t) = E(0), Note that as R → ∞, due to lim s→∞ φ(s) = 0, we obtain By our choice of M 0 , this last inequality is violated. Theorem 2.3: To get a uniform upper bound on h, note that at a maximum of h we have The right-hand side will be negative if h max → ∞. To get a lower bound on h away from zero, we can argue similarly.
In the following two lemmas, we will use two auxiliary functions from [KS19, Iva19a] to find uniform lower and upper bounds on the Gauss curvature.
Lemma 3.4. We have σ n ≥ c for some positive constant.
Proof. The evolution equation of χ := log(Θσ n ) − A ρ 2 2 is given by Dropping some positive terms gives ηρ 2 − (n + 1)AhΘσ n − Aσ nḡ ij∇ i h∇ j Θ. By the C 0 -estimate, if A is chosen large enough, the right-hand side of the above equation is strictly positive provided min σ n → 0. Thus σ n is uniformly bounded below away from zero.
Lemma 3.5. There is a constant d such that σ n ≤ d.
Proof. For ε > 0 sufficiently small, consider the auxiliary function We have Therefore, by Lemma 3.1 and (3.7), at any point with t > 0 where χ attains a maximum, = h∇ j h +ḡ kl∇ j∇k h∇ l h =ḡ kl r jl∇k h. and from the convexity of the hypersurface, we then see that the term σ ij n∇ i log h∇ j Θσn h is non-positive. Thus due to AM-GM inequality, σ ij n r k i r jk ≥ cσ 1+ 1 n n , and the C 0 -estimate, we obtain ∂ t χ ≤ c 1 χ + c 2 χ 2 − c 3 χ 2+ 1 n for some positive constants c 1 , c 2 , c 3 . Here we identified χ with max χ. From this the claim follows. 3.2. Convergence. In the previous section, the C 2 -estimates were obtained for either flow under their corresponding assumptions. Now the higher order regularity estimates follow from the theory of parabolic differential equations; see for details [McC05,Mak13]. Therefore, the maximal time interval is unbounded.
Due to Lemma 3.2, subconvergence for the flows (2.1) and (2.3) to stationary solutions is standard: by the upper bound on the support functions, there is a constant C, depending only on M 0 , such that S n hσ n dθ ≤ C.
Hence, the flow hypersurfaces subconverge to a stationary solution.
To obtain full convergence for the flow (2.3), let us put Recall from (3.5) that Note that .
Hence due to our C 0 -estimates, Now we can argue as in [And97,GW03] to promote subconvergence to full convergence.

General measures
4.1. Notions and notation. Let φ : [0, ∞) → [0, ∞) be an increasing function, continuously differentiable on (0, ∞) with positive derivative, and satisfying lim s→∞ φ(s) = ∞. For a finite non-zero Borel measure µ and a continuous, nonnegative function f on S n , the Orlicz norm f φ,µ is defined by Here |µ| = µ(S n ). Note that in general the Orlicz norm does not satisfy a triangle inequality and the case φ(t) = t p gives the normalized L p norm. The Orlicz norm satisfies the following properties: Definition 4.1.

Parabolic Approximation.
We begin this section by sketching the proof of Theorem 2.7. Assume ϕ is smooth.
Step 1: We perturb ϕ to ϕ ε such that ϕ ε (s) = s −n−ε , ∀s ∈ (0, ε]. Step 2: Suppose 0 < f ∈ C ∞ (S n ). Using a suitable curvature flow, we find a smooth, strictly convex hypersurface M ε with positive support function such that Step 3: Take ε = 1/i and set ϕ i = ϕ ε . Applying Step 2 to f and ϕ i , we find K i (with the origin in its interior) such that Moreover, we show that the minimum and maximum width of K i as well as γ i are uniformly bounded above and below away from zero, such that these bounds, for i sufficiently large, depend only on ϕ and µ f . Thus, letting i → ∞, we can find a convex body K ∈ K o such that dµ f for a constant γ > 0.
Step 4: Choose 0 < f i ∈ C ∞ (S n ) such that µ f i → µ weakly as i → ∞. By the conclusion of Step 3, we find K i ∈ K o such that Moreover, w ± (K i ) and γ i are uniformly bounded above and below away from zero, and these bounds, for i sufficiently large, depend only on ϕ and µ. Thus, letting i → ∞, we find K ∈ K o solving dµ for a constant γ > 0.
Lemma 4.4. Let µ be a finite Borel measure whose support is not contained in a hemisphere. Then for ε sufficiently small we have Due to (4.5), φ ε ≥ φ. Therefore, we have Since φ is increasing, if c ε i ≤ c 2 for a subsequence of {ε i } → 0, then
Lemma 4.5. Let a, b be two positive constants, ϕ, ϕ ε be defined as above, andK Suppose µ, {µ i } i∈N are finite Borel measures on S n such that µ i → µ weakly and the support of µ is not contained in a closed hemisphere. Then there exist positive constants λ, Λ depending on a, b, ϕ, µ such that for all K ∈K o , i sufficiently large and ε small enough, Proof. Recall that s/ϕ(s) is uniformly continuous Second, we show that {c i } is uniformly bounded below away from zero. Otherwise, we can find a sequence {K i } ⊂K o such that dµ.
In view of (4.8), each line of the right-hand side goes to zero thus yielding a contradiction. Finally, due to (4.8), for every δ > 0, there exist ε 0 , N, such that for all ε ≤ ε 0 , i ≥ N and K ∈K o , we havê Therefore, the uniform lower bound follows by taking δ small enough. Now we prove the upper bounds. For i sufficiently large, we havê Moreover, due to (4.8), there exist ε 0 , N, such that for all ε ≤ ε 0 , i ≥ N and K ∈K o , we havê Proposition 4.6. Suppose ϕ : (0, ∞) → (0, ∞) is a smooth such that Assumption 4.1 is satisfied. Let f : S n → (0, ∞) be smooth and ε be sufficiently small. There exists a closed, smooth, strictly convex hypersurface M ε with positive support function such that In addition, we have Moreover, for some positive constants λ, Λ depending on ϕ, µ f , Also, if f is invariant under a closed group G ⊂ O(n + 1), then M ε can be chosen to be G-invariant.
Proof. Let M 0 be the unit sphere. We seek a family of smooth, strictly convex hypersurfaces {M t,ε } whose support functions satisfy We prove the flow hypersurfaces subconverge smoothly to a solution of (4.9) with the desired properties. To do this, all we need is to obtain C 0 -estimates; the higher order regularity estimates and convergence follow as in Sections 3 and 3.2. Moreover, the statement regarding G-invariance follows, since the flow hypersurfaces are G-invariant.
Suppose T ⋆ is the maximal time that the flow hypersurfaces are smooth, strictly convex and contain the origin in their interiors. First, we obtain a uniform upper bound for support functions on [0, T ⋆ ) independent of T ⋆ . Then using this bound, we establish a uniform positive lower bound on the support functions on [0, T ⋆ ) independent of T ⋆ . Thus the flow hypersurfaces will always contain the origin in their interiors as long as they are smooth and strictly convex.
Let us put h t,ε = h Mt,ε and define Thus from the definition of the Orlicz norm it follows that h t,ε φε,µ f ≤ 1.
Choose v ∈ S n and R > 0, such that Rv ∈ M t,ε and |Rv| is maximal. The line segment joining the origin and Rv is the largest such line segment contained in the enclosed region by M t,ε and its support function is Rhv. Therefore, by this inclusion h t,ε ≥ Rhv. Due to (4.1), (4.11) R min v∈S n hv φε,µ f ≤ R hv φε,µ f ≤ h t,ε φε,µ f ≤ 1. By Lemma 4.4, for ε sufficiently small, we have min v∈S n hv φε,µ f ≥ 1 2 min v∈S n hv φ,µ f . Together with (4.11), this last inequality yields an upper bound on R = max h t,ε and the maximum width: Moreover, since the volume is non-decreasing along the flow, the lower bound on the minimum width follows as well: Thus by a special case of Lemma 4.5 (µ = µ i = dµ f ) we obtain for some positive constants. Therefore, as soon as h min ≤ ε, ∂ t log h min ≥ϕ ε (h min )h n min f min − Λ =h −ε min f min − Λ. Thus for each ε, h t,ε is uniformly bounded below away from zero.
Corollary 4.7. Suppose ϕ : (0, ∞) → (0, ∞) is a smooth function such that Assumption 4.1 is satisfied. Let f : S n → (0, ∞) be smooth. Then there exists a convex body K ∈ K o such that In addition, we have and for some positive constants depending only on ϕ, µ f , Moreover, if f is invariant under a closed group G ⊂ O(n + 1), then K can be chosen to be invariant under G.
Proof. Let ε = 1 i in Proposition 4.6 and ϕ i = ϕ ε . For each i, we have a solution K i containing the origin in its interior such that Moreover, {h K i } is uniformly bounded above, we have a uniform positive lower bound on w − (K i ), and γ i ∈ [λ, Λ]. Thus by the Blaschke selection theorem, we may find a subsequence of {K i } converging to a limit K ∈ K o while γ i → γ ∈ [λ, Λ]. Note that if f is invariant under G, then all K i and thus K can be chosen to be invariant under G.
Suppose ψ ∈ C(S n ). Note that In view of (4.7), the first line on the right-hand side tends to zero as i → ∞. Since 1/ϕ is continuous and h K i converges to h K uniformly, the term on the second line converges to zero as well. Since S K i → S K weakly (cf., [Sch13]), K satisfieŝ Suppose K i is a solution corresponding to the measure µ f i given by Corollary 4.7. By (4.12), we have uniform lower and upper bounds on w − (K i ) and w + (K i ). Thus a subsequence of {K i } converges to a convex body K ∈ K o .
Since µ f i → µ weakly, by Lemma 4.5 the constants λ, Λ of Corollary 4.7 depend only on w − , w + , ϕ, µ. Hence, K is the desired solution. A further approximation allows us to assume ϕ is merely continuous.
Proof of Theorem 2.6. We only mention the necessary changes in the argument leading to Theorem 2.7. Suppose {f i } ⊂ C ∞ (S n ) is a family of positive even functions such that µ f i → µ weakly as i → ∞. We consider the flow of smooth, strictly convex hypersurfaces {M t,i } whose support functions satisfy Note for each i, the flow hypersurfaces {M t,i } remain o-symmetric and converge smoothly to a strictly convex hypersurface M i , enclosing the region K i , which solves For i sufficiently large, the uniform upper bound for {h K i } follows from the following argument. For v ∈ S n , let hv be the support function of the line segment joining ±v. Then we have min v∈S n hv φ,µ f i > 0. Now for any v and R such that ±Rv ∈ M i with maximal distance from the origin, we have Rhv ≤ h K i . Therefore, arguing similarly as in (4.11), we deduce R min v∈S n hv φ,µ f i ≤ 1.
Since µ is a finite even Borel measure which is not concentrated on a great subsphere, by [HH12,Cor. 3.7] we have min v∈S n hv φ,µ f i ≥ δ > 0 for i large enough. Hence due to V (M i ) ≥ V (S n ), we find constants a, b such that This yields λ, Λ, depending only on a, b, such that Therefore, a subsequence of {K i } converges to a solution of (1.1).
Remark 4.8. Note that in Theorem 2.6, the condition lim s↓0 ϕ(s) = ∞ is relaxed. In this case, the lower bound on the support function follows easily as is explained in the proof of Theorem 2.6. While to prove Theorem 2.7, we use an approximation argument in which it is essential that the right-hand side of (4.7) converges to zero; cf., Corollary 4.7.

Christoffel-Minkowski type problems
Given a smooth, positive function f defined on the unit sphere, the L p -Christoffel-Minkowski problem asks for a smooth, strictly convex hypersurface with positive support function h that satisfies f h 1−p σ k = γ, for some constant γ > 0.
In this section, we solve a class of L p -Christoffel-Minkowski type problems, allowing a broader class of curvature functions in place of σ k , which can be considered as a complement of [BIS20, Thm. 3.13]. Our proof remains brief and only the necessary modifications of the proof in [Iva19a] are highlighted. Define Γ + = {(λ i ) = (λ 1 , . . . , λ n ) ∈ R n ; λ i > 0}.
Assumption 5.1. Suppose F ∈ C ∞ (Γ + ) is positive, 1-homogeneous, strictly monotone, F (1, . . . , 1) = 1, inverse concave, i.e., To prove this theorem, we use a curvature flow similar to (2.3). For a suitable smooth, strictly convex hypersurface M 0 parameterized by we seek smooth, strictly convex hypersurfaces M t satisfying x(·, 0) = x 0 (·). (5.1) Therefore, the support functions of the M t satisfy Compared to the flow [Iva19a, (1.4)], here we do not have the global term η. Moreover, since in general the problem f h 1−p F k = 1 does not have any variational structure, we cannot expect the convergence of solutions for all initial hypersurfaces. Therefore, we follow the approach of [BIS20,Ger06] to obtain the regularity estimates.
Step 1: Choose M 0 to be a sufficiently small origin-centered sphere such that f h 1−p F k | M 0 > 1. This is possible due to p > k + 1.
Step 2: We obtain uniform lower and upper bounds on the support function as in the proof of Lemma 3.3 here (the case of Theorem 2.3).
Step 3: Let us put Thus, f h 1−p F k is uniformly bounded above and below. By Step 2, we obtain uniform lower and upper bounds on F.
Step 4: Let r ij denote the inverse of principal radii of curvature. Since F is inverse concave, as in [Iva19a, Lem. 2.7], we can apply the maximum principle to the auxiliary function r ij h to obtain a uniform lower bound on the principal radii of curvature.
Step 6: Now that we have uniform C 2 -estimates, higher order regularity estimates follow as well and the flow smoothly exists on [0, ∞).
Therefore, in view of monotonicity of h, the limit h(u) := lim t→∞ h(u, t) exists and is positive, smooth and∇ 2h +ḡh > 0. Thus the hypersurface with support functionh is our desired solution to f h 1−p F k = 1.