A priori bounds and existence of smooth solutions to a L p Aleksandrov problem for Codazzi tensor with log-convex measure

: In the present paper, we prove the existence of smooth solutions to a L p Aleksandrov problem for Codazzi tensor with a log-convex measure in compact Riemannian manifolds ( M , g ) with positive constant sectional curvature under suitable conditions. Our proof is based on the solvability of a Monge-Amp`ere equation on ( M , g ) via the method of continuity whose crucial factor is the a priori bounds of smooth solutions to the Monge-Amp`ere equation mentioned above. It is worth mentioning that our result can be seen as an extension of the classical L p Aleksandrov problem in Euclidian space to the frame of Riemannian manifolds with weighted measures and that our result can also be seen as some attempts to get some new results on geometric analysis for Codazzi tensor.


Introduction
Let (M, g) be a n-dimensional Riemannian manifolds. In the present paper, we focus on the following Monge-Ampère equation on compact Riemannian manifolds (M, g) without boundary: for any z ∈ M where ρ 2 = g i j S i S j + S 2 and S i j + g i j S is a (0, 2) type Codazzi tensor. An equivalent form of (1.1) is S 1−p e −φ(ρ 2 ) σ n det g i j dz = ϕ(z)dz, (1.2) where σ n = λ 1 λ 2 · · · λ n and {λ l } n l=1 is a solution sequence to the following algebraic equation, det(S i j + g i j S − λg i j ) = 0. (1.3) Noting that the volume element of Riemannian manifold is dV = det g i j dz, we get S 1−p e −φ(ρ 2 ) σ n dV = ϕ(z)dz. (

1.4)
The left-side of (1.2) is called the density of p-integral Gaussian curvature measure for the log-convex measure e −φ(ρ 2 ) dx for Codazzi tensor in the present paper.
In particular, if M = S 2 , p = 1, φ ≡ 0 and S is the support function of hypersurface N ⊆ R 3 , the second fundamental form (h i j ) 2×2 of the hypersurface N is given as h i j = S i j + g i j S (1.5) for any fixed i, j ∈ {1, 2} where S i j is the covariant derivation of S of second order. Then the left hand side of (1.4) becomes σ 2 dV (1. 6) which is associated with the well-known Gauss-Bonnet formula for 2-dimensional Riemannian submanifold without boundary, see pp 358 of Kobayashi and Nomizu [1]. Lemma A.1(Gauss-Bonnet formula( [1])). Let N ⊆ R 3 be an orientable, compact, smooth 2dimensional submanifold without boundary. Then, where χ(N) ∈ Z denotes the Euler characteristic number of N.
Since the measure σ n dV has its geometric origin, and therefore it is natural to get an intrinsic construction of the measure σ n dV. In polar coordinates, one can formulate the measure σ n dV as follows: ω σ n dV = ν N (r N (ω)) dξ (1.8) for any Borel set ω ⊆ S n , where ν N and r N are the normal mapping and radial mapping of the hypersurface N, ρ N (ξ) = max{λ ≥ 0 : λξ ∈ N}, ∀ξ ∈ S n , (1.9) and r N (ξ) = ρ N (ξ)ξ, ∀ξ ∈ S n , (1.10) see Oliker [11][12][13] or Schneider [10]. This observation led Aleksandrov to pose the following classical Aleksandrov problem, see Aleksandrov [14,15], Bakelman [16] or Guan, Li and Li [17]. Problem A.2 (The classical Aleksandrov problem). For any fixed n ≥ 1, given a Borel measures µ which is supported on the unit sphere S n , finds a convex hypersurface N ⊆ R n+1 such that for any Borel set ω ⊆ S n where dξ, ν N and r N are the standard n-dimensional spherical Lebesgue measure, normal mapping and radial mapping of the hypersurface N. Aleksandrov [14,15] solved Problem A.2. via the mapping argument which is a kind of method of continuity, see also Bakelman [16] or Pogorelov [18]. Later, from the point of view of nonlinear analysis or PDEs theory, Oliker [11][12][13] resolved Problem A.2. Moreover, concerning the regularity or curvature bounds of the hypersurface, Treibergs [19] and Guan and Li [20] also analyzed the Problem A.2. Recently, Huang, Lutwak, Yang and Zhang [21] introduced the so-called p-integral Gaussian curvature measure and posed the L p Aleksandrov problem which can be stated as follows.
Problem A.3 (L p Aleksandrov problem ( [21]). For any fixed n ≥ 1 and p ∈ R, given a Borel measures µ which is supported on the unit sphere S n , find a convex hypersurface N ⊆ R n+1 such that for any Borel set ω ⊆ S n where dξ, ν N , u and r N are the standard spherical Lebesgue measure, normal mapping, support function and radial mapping of the hypersurface N.
It is interesting to focus on the geometry of weighted measure e −φ(|x| 2 ) dx without the assumption of convexity of φ.
If φ is concave, we call the measure e −φ(|x| 2 ) dx a log-convex measure. One interest in the geometry of log-convex measure e −φ(|x| 2 ) dx is the so-called log-convex density conjecture in geometric measure theory which can be stated as follows.
Problem A.4 (Log-convex density conjecture). In R n+1 with a smooth, radial, log-convex density, balls about the origin provide isoperimetric regions of any given volume.
The Log-convex density conjecture was posed by Brakke and solved by Chambers [39]. More interesting comments on this topic can be referred to [40][41][42][43].
Motivated by these beautiful results mentioned above, the main focus of the present paper is on L p Aleksandrov problem for log-convex measure e −φ(|x| 2 ) dx in the frame of Riemannian Geometry.
It is well-known that the main language of Riemannian Geometry is the so-called tensor, see Bishop and Goldberg [44] or Gerretsen [45]. This leads to our consideration on the problem in tensor spaces. By the analysis mentioned above, the core concept is the concept of Gaussian curvature. It is easy to see that the Gaussian curvature of a hypersurface can be calculated by means of the metric and second fundamental form of the hypersurface, see Kobayashi and Nomizu [1]. Therefore, to formulate a natural generalization of the classical L p Aleksandrov problem in tensor spaces, we need to replace the second fundamental form by some interesting symmetric tensors and pose a natural generalization of Gauss curvature. Since the second fundamental form of any hypersurface in a space of constant curvature satisfies the Codazzi equation, we may say a natural generalization of the second fundamental form of the hypersurface is the so-called Codazzi tensor of Riemannian manifolds in higher dimensional tensor space which is defined as follows.
For any connected smooth n-dimensional Riemannian manifolds (M, g), we let S T 2 be the bundle of smooth symmetric (0, 2) type tensor field over M, the covariant differential in the metric g is denoted by ∇ X where X is a vector field from the tangle bundle T M. The so-called Codazzi tensor is defined as follows: Definition A.5 (Cadazzi tensor [46]). Let A : M → S T 2 be a smooth section. It is called a Codazzi tensor if A satisfies Codazzi equation, for any X, Y, Z ∈ T M. The set of Codazzi tensors on M is denoted by Cod(M, g).
In particular, the second fundamental form of any hypersurface in a space of constant curvature is a Codazzi tensor, see pp. 26 of Kobayashi and Nomizu [1].
Some basic differential geometric theories about the Codazzi tensor are listed as follows, see [46]. Let x : M → R n+1 be an isometric immersion and assume also that rank A = n at z. Then rank A = n in some neighborhood U of z. Suppose that ξ is the unit normal vector field over x(U).
(i) Let x : M → R n+1 be an isometric immersion and ξ be the unit normal vector field over x(U). Then "support" function f (ξ) = −(x, ξ) where (·, ·) is the inner product in R n+1 .
(ii) the second order covariant differential of f and the coefficient of the second fundamental form Lemma A.7 ( [46]). Let (M, g) be a Riemannian manifold of constant sectional curvature K sec (possibly zero) and A ∈ Cod(M, g). Then for every point on M, there exists a neighborhood V and a smooth function f : In addition, if M is simply connected then such representation is available on the entire M. Conversely, on a manifold of constant curvature K sec , any smooth function f generates a Codazzi tensor A( f ) via Eq (1.15). For any f ∈ C ∞ (M), we let A = (A i j ) n×n be the Codazzi tensor generated by f , that is, A i j is given by (1.15). Let It is easy to see that P 0 n satisfies the following equation where λ 1 , λ 2 , · · · , λ n are n solutions to the following equation This means that the geometric meaning of P 0 n is that P 0 n is a concept of "Gaussian" curvature for the quadratic form A.
Oliker and Simon [46] proved the following interesting prescribed Gaussian curvature problem for Codazzi tensor: Theorem A.9 ( [46]). Let (M, g) be a closed Riemannian manifold with constant sectional curvature K sec 0 , and ϕ : M → (0, ∞) is a strictly positive C ∞ function. If K sec > 0 suppose also that M is not isometrically diffeomorphic to the sphere S n of R n+1 . Then there exists a (unique) function f ∈ C ∞ (M) such that is a positive definite Codazzi tensor on M and Suppose the sectional curvature K sec of M is positive constant, with loss of generality, we may assume that K sec = 1. In (1.5), if we replace the standard Lebesgue dx with e −φ(|x| 2 ) dx, the p-integral Gaussian curvature function of Codazzi tensor with log-convex measure e −φ(|x| 2 ) dx can be defined as follows: (1.20) Therefore, we let and call P n,p ( f ) as p-integral Gaussian curvature function for Codazzi tensor with log-convex measure e −φ(|x| 2 ) dx.
In the present paper, we focus on L p Aleksandrov problem for Codazzi tensor with log-convex measures e −φ(|x| 2 ) dx which is stated as follows: Problem A.10 (L p Minkowski problem for Codazzi tensor with log-convex measure). For any fixed n ≥ 1 and p ∈ R, does there exist a Codazzi tensor A whose sectional curvature is 1 and is generated by f such that P n,p ( f ) = ϕ?
The main result of the present paper can be stated as follows. Theorem 1.1. For any fixed n ≥ 1 and p > n + 1, there exist positive constants c, τ and a positive solution S ∈ C 2,τ (M) to the Eq (1.1) satisfying where τ ∈ (0, 1), c is independent of S provided the following conditions hold.
The main result of the present paper can be seen as an attempt at some new results on the integral geometry of differential forms which are associated with some interesting invariants arising in geometry and topology. Apart from the beautiful Gauss-Bonnet Theorem, there are many famous theorems which link the analysis, topology and geometry, such as Euler-Poincaré characteristic formula, Riemann-Roch-Hirzebruch-Grothendieck Theorem, Atiyah-Singer index Theorem, Chern-Simons invariants, see Palais [47], Chern-Simons [48], Atiyah [49], Shanahan [50], Mukherjee [51], Moore [52], Gilkey [53], Freed [54] and so on. Following some classical ideas of Steiner, Federer and Chern, it may be interesting to focus on some kinematic formulas for these invariants and consider analogous geometric problems which prescribe differential forms for these invariants. In particular, in the frame of Kähler Manifolds or Symplectic Manifolds, some great results on this topic can be referred to the great works of Patodi [55], Duistermaat [56], Atiyah and Bott [57], Karshon and Tolman [58], Donaldson [59][60][61], Abreu [62], Grossberg and Karshonand [63], Boyer, Calderbank and Tønnesen-Friedman [64] and so on. In our forthcoming study, we will focus on these topics.
Our proof of Theorem 1.1 is based on the well-known continuous method. We let the set of the positive continuous function on M be C + (M) and The main ingredient is the a priori bounds of solutions to the following auxiliary problem for any S ∈ C: for t ∈ [0, 1]. Remark 1.3. It is worth mentioning that without the assumption of convexity of φ, some necessary geometric inequalities have not been established which does not guarantee the validity of the classical variational framework for Problem A.10. Therefore, we adopt the well-known continuous method to solve this problem. Moreover, by the a priori bounds (1.23) of S , we get the compactness of the solution set and the curvature estimate of the Codazzi tensor which have their independent interests. Remark 1.4. Just like the case of concavity, log-convexity may be defined in the form of Prékopa-Leindler inequality: a function (or functional) f : I → R is called log-convex if f satisfies the following condition, for any t ∈ [0, 1] and x, y ∈ I. The former case can be referred to pp. 369-374 of Schneider [10]. Some inequalities for log-convex functions (or functionals) have been analyzed in [65][66][67][68] and so on. It may be worth mentioning that Klartag [22] and Rotem [28] introduced some geometric notions for log-concave or more general α-concave functions and measures, such as the support function and mean width. Motivated by these interesting results, it may be interesting to introduce similar notions and get more geometric analysis for log-convex measures and this will also be a topic of our future study. The remaining part of this paper is arranged as follows: In Section 2, we prove the a priori bounds of S . In Section 3, we prove Theorem 1.1.

A priori bounds of S
In this section, we consider the a priori bounds of solutions to the following equation on Riemannian manifolds (M, g): where ρ 2 = |∇S | 2 + S 2 and the following conditions hold. (A.1.) 0 < ϕ ∈ C 4 (M), φ is a non-negative, radially symmetric, increasing, smooth and convex function in R, 0 < ϕ ∈ C 4 (M) and We
Step (a). For any fixed n ≥ 1 and p > n + 1, we let S ∈ C ∩ C + (M) be a solution to (2.1). Suppose that (A.1) ∼ (A.3) hold. Then there exists a positive constant c such that for any z ∈ M. Proof of Step (a). We consider the following extremal problem, It follows from the compactness of M and the continuity of S that there exists z 1 ∈ M such that R = S (z 1 ).

(2.5)
It follows from the Eq (2.1) that at the point z = z 1 , Combining this and condition (A.1), we can see that there exists a positive constant c > 0 such that We next consider the following extremal problem, Adopting a similar argument mentoned above, we also can see that there exists a positive constant c > 0 such that r ≥ c > 0. (2.9) (2.7) and (2.9) yield the desired conclusion of Step (a). Step for any fixed l ∈ {1, 2, · · · , n} at the point z 0 . It follows from Lemma 2.1 that there exists a positive constant c, det(S il + S g il ) = S p−1 ϕ(z)e φ(ρ 2 ) det g il ≥ S p−1 ϕ(z)e φ(0) det g il ≥ c 0 det g il (2.14) at the point z = z 0 where c 0 = e φ(0) min z∈M S p−1 (z)ϕ(z) > 0. (2.15) Noting that g = (g i j ) n×n is strictly positive, it follows from (2.14) and (2.15) that the matric (S il +S g il ) n×n is reversible at the point z = z 0 and therefore, at the point z = z 0 for any fixed l ∈ {1, 2, · · · , n} due to (2.13) and the positivity of g. From (2.16), we can see that |∇S | 2 = g i j S i S j = 0 (2.17) at the point z = z 0 . Therefore, it follows from Lemma 2.1 that there exists a positive constant c, independent of S , for any z ∈ M. This completes the proof of Step (b). Before getting the higher order estimates of S , we let and Then, Eq (2.1) becomes G(u i j ) = Ψ. (2.21) Step (c). For any fixed n ≥ 1 and p > n + 1, we let S ∈ C ∩ C + (M) be a solution to (2.1) and u ∈ C ∩ C + (M) be a solution to (2.21). and (H i j ) n×n is non-positive. We let for any fixed i, j, s, t ∈ {1, 2, · · · , n}. Therefore, at the point z = z 0 , By the commutator identity, we have, Putting (2.25) into (2.26), we get Taking the α-th partial derivatives on both sides of (2.21) twice for any fixed α ∈ {1, 2, · · · , n}, we have Σ i j G i j u i jα = Ψ α , Σ i jst G i j,rs u i jα u rsα + Σ i j G i j (u αα ) i j = Ψ αα (2.28) for any fixed α ∈ {1, 2, · · · , n}. By the concavity of G, we have Σ i jstα G i j,rs u i jα u rsα ≤ 0. (2.29) This implies that Σ i G ii ∆u ii ≥ Σ i j G i j ∆u i j + Σ i jstα G i j,rs u i jα u rsα = ∆Ψ. (2.30) at the point z = z 0 . Therefore, at the point z = z 0 . It follows from Newton-MacLaurin inequality that We let It follows from Lemmas 2.1 and 2.2 that Now, we claim that at the point z = z 0 , where δ 4 > 0 to be chosen. Indeed, it follows from the definition of Ψ that (2.37) Noting ρ 2 = |∇S | 2 + S 2 , for any fixed α ∈ {1, 2, · · · , n}, taking α-th partial derivatives on both sides of (2.37) twice, we have (2.39) where and We now get some estimates of I 2 . Since ϕ ∈ C 4 (M), it follows from Lemmas 2.1 and 2.2 that Therefore, it follows from Lemma 2.1 and Hölder inequality that for some c which means that Moreover, it follows from (2.43), (2.23) and Lemma 2.2 that at the point z = z 0 . Therefore, combining (2.42), (2.45) and (2.46), we have, for any z ∈ M. Noting that Therefore, We let r 1 = min z∈M ρ(z), R 1 = max z∈M ρ(z). (2.51)
(2.53) Therefore, Therefore, (2.47) and (2.54) yield at the point z = z 0 . This is the desired inequality (2.36). It follows from (2.33) and (2.36) that at the point z = z 0 . It follows from Hölder inequality that at the point z = z 0 . Combining (2.57), the definition of u and Step (a), it is easy to get the inequality (2.22) which completes the proof of Step (c).
Step (d). It follows from (2.21) that Eq (2.1) becomes Step (a), Step (b) and Step (c) that there exist positive constants λ and Λ, independent of S , such that and for any ζ = (ζ 1 , ζ 2 , · · · , ζ n ) ∈ R n . That is, (d.i) (2.58) is elliptic uniformly. Now, we claim that (d.ii) F is concave with respect to (u i j ) n×n . It follows from the definition of F that it suffices to prove that G = det 1 n is concave with respect to (u i j ) n×n . Indeed, For any u, v ∈ C and t ∈ [0, 1], we let {δ 1 i } n i=1 and {δ 2 i } n i=1 be the eigenvalue sequence of (u i j ) n×n and (v i j ) n×n respectively. Then is a eigenvalue sequence of (tu i j + (1 − t)v i j ) n×n . Moreover, since (u i j ) n×n and (v i j ) n×n are convex, we have for any fixed i ∈ {1, 2, · · · , n} and j ∈ {1, 2} and therefore, for any t ∈ [0, 1]. Combining the definition of G and (2.62), we get which proves that G = det 1 n is concave with respect to (u i j ) n×n and this completes the proof of the claim. Then, it follows from (d.i), (d.ii) and Theorem 17.14 of Gilbarg and Trudinger [70] that there exist τ 1 ∈ (0, 1) and positive constant c, independent of S , such that (see pp. 457-461 of Gilbarg and Trudinger [70]). Therefore there exist τ ∈ (0, 1) and positive constant c, independent of S , such that This is the desired conclusion of Theorem 2.0.
We let the set of the positive continuous function on the Riemannian manifolds (M, g) be C + (M) and C = {S ∈ C 2,τ (M) : (S i j + S g i j ) n×n is positive definite} Since f t is a contious function, independent of z, satisfying for any t ∈ [0, 1] and z ∈ M, adopting some similar arguments in Section 2, we get Lemma 3.1. For any fixed n ≥ 1, p > n + 1 and t ∈ [0, 1], we let S t ∈ C ∩ C + (M) be a solution to (3.1). Suppose that (A.1) ∼ (A.3) hold. Then there exists a constant c, independent of t, such that for any t ∈ [0, 1] and some τ ∈ (0, 1).
As a corollary of Lemma 3.1, we have, Corollary 3.2. For any fixed n ≥ 1, p > n + 1, we let I is the set defined in (3.2). Suppose that (A.1) ∼ (A.3) hold. Then I is closed.
Proof. It suffices to show that for any sequence {t j } ∞ j=1 ⊆ I satisfying as j → ∞ for some t 0 ∈ [0, 1], we need to prove t 0 ∈ I. We let S j be a solutions of problem (3.1) at t = t j . It follows from the conclusion of Lemma 3.1 that there exists a positive constant c, independent of j such that it follows from Ascoli-Arzela Theorem that up to a subsequence, there exists a S 0 ∈ C 2 (M), such that uniformly on M as j → ∞ where (ρ j ) 2 = (S j ) 2 + |∇S j | 2 for any j ∈ {0, · · · }. Letting j → ∞, we can see that (t 0 , S 0 ) is a solution to the following problem: for any z ∈ M. (3.4) implies that t 0 ∈ I. This is the desired conclusion of Corollary 3.2. Lemma 3.3. For any fixed n ≥ 1, p > n + 1, we let I is the set defined in (3.2). Suppose that (A.1) ∼ (A.4) hold. Then I is open.
Proof. Suppose that there exists at ∈ I, it suffices to prove t ∈ I for any t ∈ B δ (t) ∩ [0, 1]. To achieve this goal, joint with Implicit Function Theorem, we need to analyze the kernel of linearized equation associated to (3.1). We assume thatS is a solution to (3.1) at t =t. For any ζ ∈ M, we let Taking logarithm on both sides of (3.8), since f t is independent ofS , we get, where (P i j ) n×n is the inverse of the matrix (S i j +S g i j ) n×n and B(ζ) = ζ i j + ζg i j . (3.10) We let ζ =S v. Direct Calculation shows that and ζ i j =S v i j + (S i v j +S j v i ) +S i j v.
Final proof of Theorem 1.1. It is easy to see that S ≡ 1 is a solution of (3.1) at t = 0. This means that I is not-empty. This, together with Corollary 3.2 and Lemma 3.3, implies that I = [0, 1]. Taking t = 1, we get the proof of Theorem 1.1.