On the uniqueness of Lp-Minkowski problems: The constant p-curvature case in R^3

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1 University of Wollongong Research Online Faclty of Engineering and Information Sciences - Papers: Part A Faclty of Engineering and Information Sciences 2015 On the niqeness of Lp-Minkowski problems: The constant p-crvatre case in R^3 Yong Hang Hnan University, hangyong@hn.ed.cn Jiakn Li University of Wollongong, jiaknl@ow.ed.a L X Hnan University, xl@hn.ed.cn Pblication Details Hang, Y., Li, J. & X, L. (2015). On the niqeness of Lp-Minkowski problems: The constant p-crvatre case in R^3. Advances in Mathematics, Research Online is the open access instittional repository for the University of Wollongong. For frther information contact the UOW Library: research-pbs@ow.ed.a

2 On the niqeness of Lp-Minkowski problems: The constant p-crvatre case in R^3 Abstract We stdy the C4 smooth convex bodies K Rn+1 satisfying K(x) =(x)1 p, where x Sn, K is the Gass crvatre of K, is the spport fnction of K, and p is a constant. In the case of n =2, either when p 1, 0] or when p (0, 1) in addition to a pinching condition, we show that K mst be the nit ball. This partially answers a conjectre of Ltwak, Yang, and Zhang abot the niqeness of the Lp-Minkowski problem in R3. Moreover, we give an explicit pinching constant depending only on p when p (0, 1). Keywords crvatre, p, constant, 3, problems, r, minkowski, lp, niqeness, case Disciplines Engineering Science and Technology Stdies Pblication Details Hang, Y., Li, J. & X, L. (2015). On the niqeness of Lp-Minkowski problems: The constant p-crvatre case in R^3. Advances in Mathematics, This jornal article is available at Research Online:

3 ON THE UNIQUENESS OF L p -MINKOWSKI PROBLEMS: THE CONSTANT p-curvature CASE IN R 3 YONG HUANG, JIAKUN LIU, AND LU XU Abstract. We stdy the C 4 smooth convex bodies K R n+1 satisfying K(x) = (x) 1 p, where x S n, K is the Gass crvatre of K, is the spport fnction of K, and p is a constant. In the case of n = 2, either when p 1, 0] or when p (0, 1) in addition to a pinching condition, we show that K mst be the nit ball. This partially answers a conjectre of Ltwak, Yang, and Zhang abot the niqeness of the L p-minkowski problem in R 3. Moreover, we give an explicit pinching constant depending only on p when p (0, 1). 1. Introdction The L p -Minkowski problem introdced by Ltwak 30] is a generalisation of the classical Minkowski problem and has been intensively stdied in recent decades. In the meantime, the classical Brnn-Minkowski theory has also been remarkably extended by Ltwak 30,31] to the Brnn-Minkowski-Firey theory. Many interesting applications and ineqalities have been correspondingly established following this pioneering development in convex geometry, see 5,10,34 36,46 48] for example. Among many excellent references, we refer the reader to the newly expanded book 40] of Schneider for a comprehensive introdction on the related topics. Yet there are still plenty of nsolved problems in this research area. In particlar, very little is known abot the niqeness of the L p -Minkowski problem when p < 1. Even in the case of n = 2, the niqeness is a very difficlt and challenging problem, and has not been settled. The aim of this paper is to establish the niqeness in R 3 for a range of p less than 1, which gives a partial answer to a conjectre of Ltwak, Yang, and Zhang abot the niqeness of the L p -Minkowski problem. Given a Borel measre m on the nit sphere S n, the L p -Minkowski problem investigates the existence of a niqe convex body K in R n+1 sch that m is the L p -srface area measre of K, or eqivalently (1.1) dm = 1 p dµ, Date: Agst 19, Mathematics Sbject Classification. Primary 35J96, 35B53; Secondary 53A05, 52A15. Key words and phrases. Minkowski problem, niqeness, Monge-Ampère eqation. Research of Hang was spported by NSFC No , Research of Li was spported by the Astralian Research Concil DE , Research of X was spported by NSFC No

4 2 YONG HUANG, JIAKUN LIU, AND LU XU where µ is the ordinary srface area measre of K and : S n R is the spport fnction of K. Obviosly, when p = 1, the L p -Minkowski problem redces to the classical Minkowski problem. We remark that when p 1, the Brnn-Minkowski-Firey theory is not a translation-invariant theory, and all convex bodies to which this theory is applied mst have the origin in their interiors. Throghot this paper, we will always assme that the origin is contained inside the interior of K, in other words, the spport fnction > 0 is strictly positive on S n. When f = dm/dx is a positive continos fnction on S n and the bondary K is in a smooth category, for example C 4 smooth, (1.1) can be described by the following Monge-Ampère type eqation: (1.2) det ( ij + δ ij ) = f p 1 on S n where ij is the covariant derivative of with respect to an orthonormal frame on S n. The case of p = 1 has been intensively stdied and landmark contribtions on reglarity are de to Lewy 28], Nirenberg 37], Calabi 9], Cheng-Ya 11], Pogorelov 39], and Caffarelli 8] among many others, see 40] for more history. For p > 1, p n + 1, Ltwak 30] solved the problem (1.1) when the given measre is even. Cho-Wang 12] solved (1.2) for a general measre when p > 1. Different proofs were presented in Hg-Ltwak-Yang-Zhang 24] for p > 1. C soltion was given by Ltwak-Oliker 32] for the even case for p > 1. For the general case, C 2,α soltion was given by Cho-Wang 12] and Gan-Lin 20] independently when p n + 1. For 1 < p < n + 1, the origin may be on the bondary of the convex body of the soltion for a measre with positive smooth density, and ths the C 2,α reglarity is not desirable, see 12, 20, 24] for an example. However, for p > 1, it was shown in Hg- Ltwak-Yang-Zhang 24] that the origin is always in the interior of the polytope of the soltion for the discrete case. The weak soltion of (1.2) for n 1 < p < n + 1 was also established and partial reglarities were obtained in 12]. For p = 0, named the logarithmic Minkowski problem (1.1), Böröczky-Ltwak-Yang-Zhang 5] obtained the existence of the even logarithmic Minkowski problem provided that the given measre satisfied the sbspace concentration condition. In the discrete case, Zh 46] dropped the evenness assmption. Recently, L-Wang 29] established the existence of rotationally symmetric soltions of (1.2) in the critical case p = n 1, see also 25,48]. By adding a gradient condition on f, Hang-L 23] obtained the C reglarity of the soltion of (1.2) for 2 < p < n + 1. The focs of this paper is on the niqeness of the L p -Minkowski problem, namely the niqeness of soltion of eqations (1.1) and (1.2). Recall that the tool sed to establish niqeness in the classical Minkowski problem is the Brnn-Minkowski ineqality (among several eqivalent forms Gardner 18]): For any convex bodies K, L R n+1 and λ (0, 1), (1.3) V ((1 λ)k + λl) V (K) 1 λ V (L) λ,

5 ON THE UNIQUENESS OF L p-minkowski PROBLEMS 3 with eqality if and only if K and L are translates, where V ( ) is the volme and + is the Minkowski sm. The niqeness of the L p -Minkowski problem for p > 1 was obtained in 30] by sing the Brnn-Minkowski-Firey ineqality: For any convex bodies K, L R n+1 containing the origin in their interiors and λ (0, 1), (1.4) V ((1 λ) K + p λ L) V (K) 1 λ V (L) λ, with eqality if and only if K = L, where + p is the Firey L p -sm and is the Firey scalar mltiplication (see Section 2 for the definitions). However, the ineqality (1.4) does not hold when p < 1 as shown in Example 2.1. The lack of sch an important ingredient cases the niqeness a very difficlt and challenging problem for the case of p < 1. Very recently, Jian-L-Wang 26] proved that for any n 1 < p < 0, there exists f > 0, C (S n ) sch that the eqation (1.2) admits two different soltions. Hence, to stdy the niqeness of the L p -Minkowski problem for p < 1, one needs to impose more conditions on the convex body K or on the fnction f. In the case of n = 1, for 0 p < 1, Böröczky- Ltwak-Yang-Zhang 6] obtained the analogos ineqalities to (1.4) for origin-symmetric convex bodies, which frther implies the niqeness nder these assmptions. When p = 0, the niqeness was de to Gage 17] within the class of origin-symmetric plane convex bodies that are also smooth and have positive crvatre; while when the plane convex bodies are polytopes, the niqeness was obtained by Stanc 42]. As mentioned in 6]: For plane convex bodies that are not origin-symmetric, the niqeness problem (when 0 p < 1) remains both open and important. On the other hand, one can ask for the niqeness when f is a positive constant in (1.2). Note that when p < 1, whether the soltion convex body K is origin-symmetric appears to be an open problem Ltwak 30], even for the special case of n = 2 and p = 0, which was conjectred by Firey 14]. Concerning the niqeness in the smooth category, bt withot the origin-symmetric assmption, the following conjectre has been posed by Ltwak, Yang, and Zhang. A conjectre of Ltwak-Yang-Zhang: Let K be a C 4 smooth convex body in R n+1 containing the origin in its interior. Let be the spport fnction of K, and K(x) be the Gass-Kronecker crvatre at the point of K with the nit oter normal x S n. When n 1 < p < 1, if the p-crvatre fnction of K is a positive constant, i.e. (1.5) (x) 1 p K(x) = C x S n, then K mst be a ball. In other words, if C 4 (S n ) is a positive soltion of (1.2) with f = C a positive constant, then mst be a constant on S n. In that case, = C 1/(n+1 p) and K is a ball of radis, centred at the origin.

6 4 YONG HUANG, JIAKUN LIU, AND LU XU The above index n 1 is critical in the sense that when p = n 1, the eqation (1.2) becomes invariant nder all projective transformations on S n, and when f is a positive constant, it is well known that all ellipsoids centred at the origin with the constant affine distance are soltions, see for example 27, 38, 43]. Withot loss of generality, by a rescaling we may assme that the constant C = 1 in eqation (1.5). Under some appropriate conditions, in the following we shall prove that = 1 is the niqe soltion of the Monge-Ampère eqation (1.6) det ( ij + δ ij ) = p 1 on S 2, which correspondingly answers the conjectre of Ltwak-Yang-Zhang in R 3. Theorem 1.1. The conjectre of Ltwak-Yang-Zhang holds tre in R 3 nder either of the following two conditions: (i) 1 p 0; (ii) 0 < p < 1 and the bondary K satisfies a pinching relation that κ 1 κ 2 β(p)κ 1, where κ 1, κ 2 are two principal crvatres. In particlar, the pinching constant is explicitly given by (1.7) β(p) = q 2 1, where q = 1 p. 1 q 2 As mentioned before, de to the lack of ineqality (1.4) in the case of p < 1, one need different tools and new ideas to stdy the niqeness of the L p -Minkowski problem. In this paper, we shall work with the Monge-Ampère type eqation (1.6) and se a maximm principle argment to prove Theorem 1.1. We remark that the Monge-Ampère type eqation (1.6) is related to the homothetic soltions of powered Gass crvatre flows X (1.8) t = Kα ν, which has been stdied by many people. In the case of n = 1, a complete classification for the homothetic soltions of crve flows was given by Andrews 3]. In higher dimensions, the classification for the homothetic soltions remains an open qestion 21]. One may conslt Andrews 1 3], Urbas 44, 45], and the references therein for related works in this direction. Or proof of Theorem 1.1 was in fact inspired by these works, and particlarly the recent paper of Andrews-Chen 4]. Theorem 1.1 (i) corresponds to the work of Andrews-Chen on the srface flows (1.8) where n = 2 and α 1/2, 1]. However, their proof depends on previos known reslts at the two end points α = 1/2 by Chow 13] and α = 1 by Andrews 1]. In this paper we give a straightforward and self-contained proof to the niqeness of (1.5) and (1.6).

7 ON THE UNIQUENESS OF L p-minkowski PROBLEMS 5 Theorem 1.1 (ii) corresponds to the work of Chow 13] that for α 1/n, there exist constants 0 C(α) 1/n depending continosly on α with C(1/n) = 0 and lim α C(α) = 1/n sch that if the initial hypersrface satisfies h ij C(α)Hg ij (where g ij, h ij, H are the 1 st, 2 nd fndamental forms and the mean crvatre respectively), then by a rescaling the limit soltion converges to a sphere. However, no explicit expression for sch a pinching constant was given by Chow. In this paper, we derive the constant (1.7) in the case of n = 2, which matches Chow s asymptotic conditions by observing that α = 1 β(p), C(α) = 1 p 1 + β(p). Moreover, when n = 2, or reslt implies that Chow s pinching constant C(α) = 0 for all α 1/2, 1] and lim α C(α) = 1/2. The organisation of the paper is as follows. In Section 2, we recall some basic facts and notions in convex geometry and differential geometry, which will be sed in or sbseqent calclations. In Section 3, we give the proof of the main theorem, which is divided into three cases p 1, 0), p = 0, and p (0, 1). The first and last cases are proved via a nified formla derived by a maximm principle argment, while the case p = 0 is de to the strong maximm principle. Concerning the flow eqation (1.8), Franzen 16] recently pointed ot that maximm-principle fnctions for any power α larger than one of the Gass crvatre does not exist, which makes it reasonable to assme the pinching condition in Theorem 1.1 (ii) for 0 < p < 1. For the remaining case that 3 < p < 1, the crrent method does not work de to a technical obstrction, and we decide to treat it in a separate paper. The corresponding qestion in Gass crvatre flows (1.8) that whether a closed strictly convex srface converges to a rond point for powers 1 4 < α < 1 2 is still open. However, for powers 0 < α < 1 4 the existence of non-spherical homothetic soltions of (1.8) have been constrcted by Andrews 2], which implies that there is no niqeness for problems (1.5) and (1.6) when p < 3. Last, p = 3 is the critical case that all ellipsoids centred at the origin with constant affine distance are soltions of (1.5) and (1.6). 2. Preliminaries 2.1. Basics of convex geometry. We briefly recall some notations and basic facts in convex geometry. For a comprehensive reference, the reader is referred to the book of Schneider 40]. A convex body K in R n+1 is a compact convex set that has a non-empty interior. The spport fnction K : R n+1 R associated with the convex body K is defined, for x R n+1, by (2.1) K (x) = max{ x, y : y K},

8 6 YONG HUANG, JIAKUN LIU, AND LU XU where x, y is the standard inner prodct of the vectors x, y R n+1. One can see that the spport fnction is positively homogeneos of degree one and convex, ths it is completely determined by its vale on the nit sphere S n. It is well known that there is a one-to-one correspondence between the set of all convex bodies, K, in R n+1 and the set S whose members are the fnctions C(S n ) sch that is convex after being extended as a fnction of homogeneos degree one in R n+1. If S, it can be shown that is the spport fnction of a niqe convex body K given by (2.2) K = x S n {y R n+1 : x, y (x)}. A basic concept in the classical Brnn-Minkowski theory is the Minkowski combination λk + λ L of two convex bodies K, L and two constants λ, λ > 0, given by an intersection of half-spaces, (2.3) λk + λ L = x S n {y R n+1 : x, y λ K (x) + λ L (x)}, where K, L are the spport fnctions of K, L respectively. The combination (2.3) was generalised by Firey 15] to the L p -combination for p 1, (2.4) λ K + p λ L = x S n {y R n+1 : x, y p λ p K (x) + λ p L (x)}, where is written for Firey scalar mltiplication. From the homogeneity of (2.4), one can see that the relationship between Firey and Minkowski scalar mltiplications is λ K = λ 1/p K. The Firey L p -combination (2.4) leads to the Brnn-Minkowski-Firey theory as developed later by Ltwak 30, 31], which has fond many applications, see for example, 33] and the references therein. Note that the Brnn-Minkowski-Firey theory is not a translation-invariant theory, and applied to the set of convex bodies containing the origin in their interiors, K 0, in R n+1. Correspondingly, we consider the set of spport fnctions S 0 = S { > 0 on S n }. Within these sets we can frther extend the Firey L p -combination (2.4) to the case of p < 1 as follows. Let a, b > 0 and 0 < λ < 1, define { (1 λ)a (2.5) M p (a, b, λ) = p + λb p ] 1/p if p 0, a 1 λ b λ if p = 0. We also define M (a, b, λ) = min{a, b}, and M (a, b, λ) = max{a, b}. These qantities and generalisations are called pth means or p-means 22]. The arithmetic and geometric means correspond to p = 1 and p = 0, respectively. Moreover, if p < q, then (2.6) M p (a, b, λ) M q (a, b, λ), with eqality if and only if a = b (as a, b > 0).

9 ON THE UNIQUENESS OF L p-minkowski PROBLEMS 7 Definition 2.1. Let K 0 be the set of convex bodies in R n+1 containing the origin in their interiors. Let K, L K 0 and K, L be the spport fnctions, respectively. For any p R, λ (0, 1), the generalised Firey L p -combination (1 λ) K + p λ L is defined by (2.7) (1 λ) K + p λ L = x S n {y R n+1 : x, y M p ( K (x), L (x), λ)}, where M p is the fnction in (2.5). As an intersection of half-spaces, the combination (2.7) gives a convex body in R n+1 for all p R. By a rescaling, one can see that when p = 1, (2.7) is the Minkowski combination (2.3), and when p > 1, it is the Firey L p -combination in (2.4). Note that when p 1, the convex body (1 λ) K + p λ L has exactly M p ( K (x), L (x), λ) as its spport fnction. However, when p < 1, the convex body (1 λ) K + p λ L is the Wlff shape of the fnction M p ( K (x), L (x), λ), which makes it very difficlt to work with 6, 40]. In particlar, the following example (as mentioned in 6]) shows that the important Brnn-Minkowski-Firey ineqality (1.4), which was a crcial tool to establish the niqeness for p 1, does not hold when p < 1 in general. Example 2.1. Let A := {x R n+1 : x i a constant. Let A ε := {x R n+1 : x 1 ε a, x j a i = 1,, n + 1}, where a > 0 is a j = 2,, n + 1}, for a small positive constant ε < a. Then A, A ε K 0. Let λ (0, 1), from Definition 2.1 (1 λ) A + p λ A ε = { M p (a, a ε, λ) x 1 M p (a, a + ε, λ)} { x j a, j > 1}. It is easy to see that V ((1 λ) A + p λ A ε ) = (2a) n (M p (a, a ε, λ) + M p (a, a + ε, λ)) and V (A) = V (A ε ) = (2a) n+1. For λ 0, 1], define h(λ) := M p (a, a ε, λ) + M p (a, a + ε, λ), where M p is in (2.5). Notice that h is a smooth fnction in λ and h(0) = h(1) = 2a for all p R. By differentiation, for λ (0, 1), h (λ) 0 if p 1, while h (λ) > 0 if p < 1. So, This implies that, for λ (0, 1), M p (a, a ε, λ) + M p (a, a + ε, λ) 2a if p 1, M p (a, a ε, λ) + M p (a, a + ε, λ) < 2a if p < 1. V ((1 λ) A + p λ A ε ) V (A) 1 λ V (A ε ) λ if p 1, V ((1 λ) A + p λ A ε ) < V (A) 1 λ V (A ε ) λ if p < 1. Following Definition 2.1, the L p -mixed volme V p (K, L) is defined by (2.8) n + 1 V (K + p ε L) V (K) V p (K, L) = lim, p ε 0 + ε

10 8 YONG HUANG, JIAKUN LIU, AND LU XU where V (K) is the volme of K. It was shown in 30] that for any K K 0, there exists a Borel measre µ p (K, ) on S n sch that the L p -mixed volme V p has the following integral representation: (2.9) V p (K, L) = 1 p L n + 1 dµ p(k, ) S n for all L K 0. The measre µ p is called the L p -srface area measre of K. When p = 1, it redces to the ordinary srface area measre µ for K. It trns ot that µ p is related to µ by 30]: (2.10) dµ p dµ = 1 p. In the smooth category when K C 2, dµ = K 1 dx, where K is the Gass crvatre of K and dx is the spherical measre on S n. In view of this, the conjectre of Ltwak- Yang-Zhang asking whether the ball is the niqe convex body sch that its L p -srface area measre dµ p eqals to the spherical measre dx on S n is eqivalent to (2.11) K = 1 p on S n. Choosing an orthonormal frame on S n, (2.11) can be written as (2.12) det ( ij + δ ij ) = 1 K = p 1 on S n. Proposition 2.1. When p 1, the nit ball 1 is the niqe soltion of (2.12), (p to a translation if p = 1). Proof. This can be proved by sing the Brnn-Minkowski-Firey ineqality (1.4) in convex geometry, even in the non-smooth category 30]. Here we reminisce some analytical reslts. When p > n + 1, consider the eqation (2.12) at the maximm max and the minimm min, one immediately has 1. In fact, Simon 41] proved that if K is smooth and satisfies (2.13) S k (κ) = G(), for a C 1 fnction G with G 0, where κ = (κ 1,, κ n ) are the principal crvatres of K, S k is the k th elementary symmetric fnction on R n, and > 0 is the spport fnction of K, then K mst be a ball. Therefore, the proposition follows as a special case of k = n, G() = 1 p, which satisfies Simon s assmption G 0 when p Basics of differential geometry. We choose a local orthonormal frame {e 1,..., e n+1 } at the position vector X K sch that e 1, e 2,, e n are tangential to K and e n+1 = ν is the nit oter normal of K at X. Covariant differentiation on K in the direction e i is denoted by i. The metric and second fndamental form of K is given by g ij = e i, e j, h ij = D ei ν, e j,

11 ON THE UNIQUENESS OF L p-minkowski PROBLEMS 9 where D denotes the sal connection of R n+1. We list some well-known fndamental formlas for the hypersrface K R n+1, where repeated indices denote smmation as the common convention. (2.14) (2.15) (2.16) (2.17) j i X = h ij ν i ν = h ij j X l h ij = j h il R ijkl = h ik h jl h il h jk (Gass formla) (Weigarten eqation) (Codazzi formla) (Gass eqation), where R ijkl is the Riemannian crvatre tensor. We also have (2.18) l k h ij = k l h ij + h mj R imlk + h im R jmlk = j i h kl + (h mj h il h ml h ij )h mk + (h mj h kl h ml h kj )h mi. Let = X, ν be the spport fnction of K. Using the above formlas, we have some identities to be sed in the next section. Lemma 2.1. For any i, j, l = 1,, n, (2.19) (2.20) i = h il l X, X, j i = h ij, X + h ij h il h jl, where h ij := k ( kh ij ) k X. Proof. Differentiating = X, ν we have i = X, i v + i X, v = X, h il l X = h il l X, X. From (2.14) (2.16) and a frther differentiation, we have j i = i X, j v + X, i j v = h il l X, j X + X, ( j h ik ) k X + X, h il j l X = h ij + h ij, X X, v h il h jl, and the proof is done. The principal crvatres λ 1, λ 2,, λ n of K are defined by the eigenvales of h ij ] with respect to the first fndamental form g ij ]. λ 1, λ 2,, λ n, S k (λ 1, λ 2,, λ n ) = The k-th elementary symmetric fnction of 1 i 1 < <i k n λ i1 λ ik, is called the k-th mean crvatre of K. In particlar, when k = 1 the mean crvatre is H = λ 1 + λ λ n, and when k = n the Gass-Kronecker crvatre is K = λ 1 λ 2 λ n. 1 If K = λ 1 λ 2 λ n 0, the reciprocals λ 1, 1 λ 2,, 1 λ n are called the radii of principal crvatre. They are eigenvales of ij + δ ij ] with respect to an orthonormal frame of S n,

12 10 YONG HUANG, JIAKUN LIU, AND LU XU where ij is the covariant derivative of on S n, and is the spport fnction. As mentioned in (2.12), we have the Gass-Kronecker crvatre (2.21) K = 1 det ( ij + δ ij ). Define the operator F (h ij ) := S n (λ(h ij )) = det h ij. Eqation (2.12) can be written as (2.22) F (h ij ) = K = q, with q := 1 p. Denote F ij = F h ij, F ij,rs = 2 F h ij h rs. Using the above formlas (2.16) (2.18), we have F ij m m h ij = F ij m i h mj (2.23) = F ij i m h mj + R miml h lj + R mijl h lm ] = F ij i j h mm + F ij h li h lj h mm F ij h ij A 2, where A 2 := m,l h2 ml. Lemma 2.2. When the dimension n = 2, we have the following: (2.24) (2.25) F ij i j H = F ij,rs m h ij m h rs + (1 q)kh 2 + (2q 4)K 2 +q q 1 H, X + q q 1 H + (1 1 q ) K 2 K, F ij i j K = q q 1 K, X + 2q q 1 K qk 2 H +(1 1 ij q )F i K j K. K Proof. Differentiating eqation (2.22) with respect to e m twice yields (2.26) F ij m h ij = m K = q q 1 m, (2.27) F ij m m h ij + F ij,rs m h ij m h rs = K = q q 1 + q(q 1) q 2 2. Using (2.23) and (2.27), we can get (2.28) F ij i j H = F ij,rs m h ij m h rs + F ij h ij A 2 F ij h im h jm H + q q 1 + q(q 1) q 2 2.

13 ON THE UNIQUENESS OF L p-minkowski PROBLEMS 11 In fact, withot loss of generality we may assme that h ij is diagonal at X. By (2.20) in Lemma 2.1 and (2.26), (2.29) F ij i j H = F ij,rs m h ij m h rs + F ij h ij A 2 F ij h im h jm H ] + q q 1 H, X + H A 2 + (q 1) 2 = F ij,rs m h ij m h rs + 2K(H 2 2K) H 2 K ] + q q 1 H, X + H (H 2 2K) + (q 1) 2 = F ij,rs m h ij m h rs + (1 q)kh 2 + (2q 4)K 2 where we have sed the 2-homogeneity of F = K. + q q 1 H, X + q q 1 H + (1 1 q ) K 2 K, On the other hand, by eqations (2.20), (2.22), and (2.26) one can obtain (2.30) F ij i j K = q(q 1) q 2 F ij i j + q q 1 F ij i j = q q 1 K, X + 2K F ij h im h mj + (q 1) F ij ] i j = q q 1 K, X + 2q q 1 K qk 2 H + (1 1 ij q )F i K j K. K 3. Proof of Theorem 1.1 In this section, we prove Theorem 1.1 by sing a maximm principle argment inspired by Andrews and Chen s work 4] on the powered Gass crvatre flow. We first derive a nified stopover ineqality (3.23), and then divide the proof of Theorem 1.1 into three cases, in three sbsections, respectively. Throghot this section, we assme the dimension n = 2. Define the axiliary fnction (3.1) Q = (λ 1 λ 2 ) 2 K α = (H 2 4K)K α, for a constant α to be determined, where λ 1, λ 2 are two principal crvatres of K; H and K are the mean and Gassian crvatres, respectively. Assme that Q attains its positive maximm vale at ˆX K. By continity, Q > 0 in a small neighborhood of ˆX. Choose an orthonormal frame sch that e 1, e 2 are tangential to K and the matrix h ij ] is diagonal at ˆX. By differentiation, we have at ˆX (3.2) 0 = i (log Q) = 2H ih 4 i K H 2 4K + α ik K,

14 12 YONG HUANG, JIAKUN LIU, AND LU XU and (3.3) 0 F ij i j (log Q) = F ij (2H i j H 4 i j K) H 2 4K + 2F ij i H j H H 2 4K + α F ij i j K K (α2 + α) F ij i K j K K 2. Or sbseqent plan is to show, however, that F ij i j (log Q) > 0 by some deliberate choice of α in (3.1). This contradiction will imply that (3.4) (λ 1 λ 2 ) 2 K α = 0 on K, namely λ 1 λ 2, and therefore, the convex body K mst be a ball. (3.5) Combining (2.24) and (2.25) into (3.3), we have From (3.2), (3.6) Hence, we obtain 0 (2 2q αq)kh + 2q(1 + α) q 1 2H H 2 4K F ij,rs m h ij m h rs ] 2H + q q 1 1q H 2 H, X + (1 ) K 2 4K K ( ) α + K 4 H 2 q q 1 K, X + (1 1 ij ] 4K q )F i K j K K + 2F ij i H j H H 2 4K (α2 + α) F ij i K j K K 2. ( ) 2H α H 2 H, X + 4K K 4 H 2 K, X = 0. 4K (3.7) 0 (2 2q αq)kh + 2q(1 + α) q 1 + L, where L are the remaining derivative terms given by (3.8) 2H L := H 2 4K F ij,rs m h ij m h rs + ( α + K 4 H 2 4K + 2F ij i H j H H 2 4K ) (1 1 q )F ij i K j K K (α2 + α) F ij i K j K K 2. 2H H 2 4K (1 1 q ) K 2 K Now, let s first estimate these derivative terms in L. diagonal at ˆX. As λ 1 = h 11, λ 2 = h 22, we have Notice that the matrix h ij ] is (3.9) i K = λ 2 i h 11 + λ 1 i h 22, F ij,rs m h ij m h rs = 2( m h 11 m h 22 m h 12 2 ).

15 Combining (3.9) into (3.8), (3.10) ON THE UNIQUENESS OF L p-minkowski PROBLEMS 13 4H L = H 2 4K ( mh 11 m h 22 m h 12 2 ) 2H + H 2 4K (1 1 q )(λ 2 1 h 11 + λ 1 1 h 22 ) 2 + (λ 2 2 h 11 + λ 1 2 h 22 ) 2 K ( ) 4K + α H 2 (1 1 ] 4K q ) λ2 1 K 2 + λ 1 2 K 2 (α2 + α) K 2 + 2λ 2 1 H 2 + 2λ 1 2 H 2 H 2. 4K We see from (3.2) and (3.8) that 2(λ 1 λ 2 )K + αλ 2 (H 2 4K)] i h 11 = 2(λ 1 λ 2 )K αλ 1 (H 2 4K)] i h 22. Since λ 1 λ 2 at ˆX, we have Denote 2K + αλ 2 (λ 1 λ 2 )] i h 11 = 2K αλ 1 (λ 1 λ 2 )] i h 22. (3.11) i h 11 = t i h 22, where t := 2K αλ 1(λ 1 λ 2 ) 2K + αλ 2 (λ 1 λ 2 ). At this stage we assme that t 0 is well defined, bt postpone the verification of this assmption in each sbseqent proof. From (2.16), (3.10), and (3.11), we then obtain 4H L = ( 1t H 2 4K 1t ] 2 ) 1h (t t 2 ) 2 h H + H 2 4K (1 1 q )(λ 2 + λ1 t ) 2 1 h (tλ 2 + λ 1 ) 2 2 h 22 2 K + 4K H 2 4K (1 1 q ) α2 α ] λ2 (λ 2 + λ 1 t ) 2 1 h λ 1 (λ 2 t + λ 1 ) 2 2 h 22 2 q K 2 + 2λ 2(1 + 1 t )2 1 h λ 1 (t + 1) 2 2 h 22 2 ] H 2 4K =: L 1 1 h L 2 2 h 22 2, where the coefficients L 1 and L 2 are respectively, (3.12) 4H L 1 = H 2 4K (1 t 1 t 2 ) + 2H H 2 4K (1 1 q )(λ 2 + λ1 K + 4K H 2 4K (1 1 q ) α2 α ] λ2 (λ 2 + λ 1 t ) 2 q K 2 + 2λ 2(1 + 1 t )2 H 2 4K t ) 2

16 14 YONG HUANG, JIAKUN LIU, AND LU XU and (3.13) 4H L 2 = H 2 4K (t 2H t2 ) + H 2 4K (1 1 q )(tλ 2 + λ 1 ) 2 K + 4K H 2 4K (1 1 q ) α2 α ] λ1 (λ 2 t + λ 1 ) 2 q K 2 + 2λ 1(t + 1) 2 H 2 4K. In order to estimate L 1 and L 2, we need some frther simplifications. In fact, by the definition of t in (3.11), we can eliminate t from L 1 as follows. (3.14) (3.15) (3.16) 1 t 1 t 2 = α(λ 1 λ 2 )H2K + αλ 2 (λ 1 λ 2 )] 2K αλ 1 (λ 1 λ 2 )] 2, (λ 2 + λ 1 t )2 = 4K 2 H 2 2K αλ 1 (λ 1 λ 2 )] 2, (1 + 1 t )2 = 4K α(λ 1 λ 2 ) 2 ] 2 2K αλ 1 (λ 1 λ 2 )] 2. Therefore, (3.17) L 1 = 4H α(λ 1 λ 2 )H2K + αλ 2 (λ 1 λ 2 )] H 2 4K 2K αλ 1 (λ 1 λ 2 )] 2 2H + H 2 4K (1 1 q ) 4KH 2 2K αλ 1 (λ 1 λ 2 )] 2 + 4K H 2 4K (1 1 q ) α2 α ] 4H 2 λ 2 q 2K αλ 1 (λ 1 λ 2 )] 2 + 2λ 2 4K α(λ 1 λ 2 ) 2 ] 2 H 2 4K 2K αλ 1 (λ 1 λ 2 )] 2, and a frther simplification gives L 1 = { 2λ 2 (H 2 4K)2K αλ 1 (λ 1 λ 2 )] 2 4H 3 (1 1 q )λ 1 8(1 1 q )KH2 + 2αH 2 2λ 2 1 2K + αh 2 4αK] 2( α q + α2 )H 2 (H 2 4K) + 4(1 + α)k αh 2 ] 2 }.

17 ON THE UNIQUENESS OF L p-minkowski PROBLEMS 15 Denote the combination in crly brackets by B 1 := (H2 4K)2K αλ 1 (λ 1 λ 2 )] 2 2λ 2 L 1. For the sake of the sbseqent analysis, let s compte B 1 in the following manner, B 1 = 2αH 2 2λ 2 1 2K + αh 2 4αK] + 4H 3 (1 1 q )λ 1 8(1 1 q )KH2 2( α q + α2 )H 2 (H 2 4K) + 4(1 + α)k αh 2 ] 2 =16(1 + α) 2 K 2 + H 2 {4αλ 2 1 4αK + 2α 2 H 2 8α 2 K + 4(1 1 q )Hλ 1 8(1 1 q )K 2(α q + α2 )H 2 + 8( α q + α2 )K 8α(1 + α)k + α 2 H 2 } { =16(1 + α) 2 K 2 + H 2 4αλ α 2 H 2 + 4(1 1 q )λ2 1 2( α q + α2 )H 2 + α 2 H 2 + K 8(1 1 q ) 8α2 4α + 4(1 1 ]} q ) + 8(α q + α2 ) 8α(1 + α), and ths (3.18) { B 1 = 16(1 + α) 2 K 2 + H 2 4(1 + α 1 q ) + α(α 2 ] q ) λ α(α 2 q )λ (α 2 2α q )K + K 4(1 1 q ) 8α2 12α + 8α ]}. q = 16(1 + α) 2 K 2 + H 2 α(α 2 q )H2 + 4(1 + α 1 q )λ2 1 4(1 + 2α)(1 + α 1 ] q )K. Regarding (3.13), we can simplify L 2 similarly as above. (3.14) (3.16) we have From (3.11), analogos to (3.19) (3.20) (3.21) t t 2 = α(λ 1 λ 2 )H2K αλ 1 (λ 1 λ 2 )] 2K + αλ 2 (λ 1 λ 2 )] 2, (λ 2 t + λ 1 ) 2 = 4K 2 H 2 2K + αλ 2 (λ 1 λ 2 )] 2, (1 + t) 2 = 4K α(λ 1 λ 2 ) 2 ] 2 2K + αλ 2 (λ 1 λ 2 )] 2. Noting the symmetry of λ 1 and λ 2 in the simplification, we then obtain L 2 =: 2λ 1 (H 2 4K)2K + αλ 2 (λ 1 λ 2 )] 2 B 2,

18 16 YONG HUANG, JIAKUN LIU, AND LU XU where (3.22) B 2 = 16(1 + α) 2 K 2 + H {α(α 2 2 q )λ (1 + α 1q ) + α(α 2q ] ) λ (α 2 2α q )K + K 4(1 1 q ) 8α2 12α + 8α ]} q = 16(1 + α) 2 K 2 + H 2 α(α 2 q )H2 + 4(1 + α 1 q )λ2 2 4(1 + 2α)(1 + α 1 ] q )K. A stopover: Finally, retrning to (3.7) and (3.8) we have the ineqality (3.23) 0 (2 2q αq)kh + 2q(1 + α) q 1 + 2λ 2 1 h 11 2 (H 2 4K)2K αλ 1 (λ 1 λ 2 )] 2 B 2λ 1 2 h (H 2 4K)2K + αλ 2 (λ 1 λ 2 )] 2 B 2, where B 1, B 2 are in (3.18) and (3.22), respectively. In the following proofs, by some deliberate choice of α we will show that the right hand side of ineqality (3.23) is positive and ths obtain (3.4) by contradiction Case I, p ( 1, 0]. Eqivalently, one has 1 q = 1 p < 2. Choosing (3.24) α = 2 q 2, we have 1 < α 0 and (3.25) (3.26) 2 2q αq = 0, 2q(1 + α) = 2(2 q) > 0. Note that in (3.11), t = (2+α)K αλ2 1. When 2 α 0, (2 + α)k αλ 2 (2+α)K αλ 2 i 2 so that t > 0 is well defined. > 0 for i = 1, 2, Then compting the crly brackets in B 1, (3.18), we have the coefficient of λ 2 1 is 4(1 + α 1 q ) + α(α 2 q ) = α + 2 (2 2q αq) = 0, q the coefficient of λ 2 2 is α(α 2 ) = 2α 0, q and the coefficient of K is (3.27) 2(α 2 2α q ) 4(1 1 q ) 8α2 12α + 8α q = 2α(2α + 3) 0, provided that α 3 2, 0]. Therefore, B 1 0, and similarly B 2 0. Hence, by (3.23) and (3.26) we obtain the contradiction (3.28) 0 2q(1 + α) q 1 = 2(2 q) q 1 > 0, which implies (3.4) and the convex body K mst be a ball.

19 ON THE UNIQUENESS OF L p-minkowski PROBLEMS Case II, p = 1. Define G = H 2. By differentiation we have (3.29) (3.30) i G = ih ij G = ijh H i 2, ih j 2 jh i 2 H ij 2 + 2H i j 3. So, (3.31) F ij ij G = F ij ij H = F ij ij H 2F ij i H j 2 HF ij ij 2 =: R + 2F ij j i G, HF ij ij 2 + 2HF ij i j ( 3 i H H ) i 2 2F ij j where R = F ij ij H HF ij ij 2. Now we compte R. From Lemmas 2.1 and 2.2, as q = 2 we have (3.32) F ij ij H = F ij,rs m h ij m h rs KH H, X + 2H + K 2 2K, (3.33) F ij ij = 2, X + 2K KH. Hence, (3.34) R = 1 F ij,rs m h ij m h rs KH2 + 2 H, X 2 H, X + 2H + 1 K 2 2 K 2HK 2 + KH2 K 2 = 1 F ij,rs m h ij m h rs K + 2 G, X. Therefore, G satisfies a niformly elliptic eqation (3.35) F ij ij G 2F ij j i G 2 G, X = 1 F ij,rs m h ij m h rs + 1 K 2 2 K =: 1 S.

20 18 YONG HUANG, JIAKUN LIU, AND LU XU From (3.9) we have (3.36) S = K 2 2K F ij,rs m h ij m h rs = λ2 2 1h λ 2 1 1h λ2 2 2h λ 2 1 2h K 2K + 1 h 11 1 h h 11 2 h 22 2( 1 h 11 1 h h 11 2 h 22 ) + 2 m h h 11 1 h h 11 2 h h 11 1 h h 11 2 h 22 2( 1 h 11 1 h h 11 2 h 22 ) + 2 m h 12 2 So we know G satisfies = 2 m h (3.37) F ij ij G 2F ij j i G 2 G, X 0. By the strong maximm principle 19], G is a constant on K. This implies that Q = (H 2 4K)K 1 is constant. Hence, from (3.23), (3.27), and comptations in the previos case, 0 F ij i j (log Q) = B 1 1 h B 2 2 h 22 2, and the coefficients B 1, B 2 > 0. So, 1 h 11 = 2 h 22 = 0, and ths by (3.11), h ij 0. Therefore, K is constant and K mst be a ball. In fact, sing the above methods in Sbsections 3.1 and 3.2, we can also obtain the following stability reslt. Corollary 3.1. For p 1, 0], if (ν) 1 p K(ν) = f > 0, then the C 4 smooth convex body K is almost a ball in the sense of (λ 1 λ 2 ) 2 C( f, 2 f ), where λ 1, λ 2 are principal crvatres of K. The constant C( f, 2 f ) = 0 provided that f is a positive constant Case III, p (0, 1). In this case, one has 0 < q = 1 p < 1. Choosing (3.38) α = 1 q 1,

21 ON THE UNIQUENESS OF L p-minkowski PROBLEMS 19 we have α > 0 and (3.39) (3.40) 2 2q αq = 1 q > 0, 2q(1 + α) = 2. By direct compting, we have B 1, B 2 in (3.23) eqal to Withot loss of generality, we assme that B 1 = B 2 = 16 q 2 K2 + (1 1 q 2 )H4. (3.41) λ 1 > λ 2 = βλ 1 for some β (0, 1). Then H = (1 + β)λ 1, K = βλ 2 1, and B 1 = B 2 = 16β 2 + (q 2 1)(1 + β) 4] λ 4 1 q 2 =: h(β) λ4 1 q 2. Let τ = 1 q 2, then h(β) = 16β 2 τ 2 (1 + β) 4. Straightforward comptations yield that h(β) 0, if β β(q), where the pinching constant β(q) is given by (3.42) β(q) = q q 2 Now, let s verify that t in (3.11) is well defined, as well the denominators in (3.23) are nonzero. Since α > 0 and λ 1 > λ 2, it sffices to show (2 + α)k > αλ 2 1. From (3.38) and (3.41), this is consistent only if (3.43) β > β t (q) := 1 q 1 + q. By direct comptation, one can see β(q) > β t (q) when q (0, 1). Hence, t > 0 is well defined when β β(q). Therefore, when β β(q), from (3.23) we have the contradiction (3.44) 0 (1 q)kh + 2 q 1 > 0, which then implies (3.4) and the convex body K mst be a ball. Acknowledgments We wold like to thank Ben Andrews for discssions on the corresponding problem of powered Gass crvatre flows.

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