Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Li, Qi-Rui; Sheng, Weimin; Wang, Xu‐Jia

doi:10.4171/jems/936

Cited by 102 publications

(69 citation statements)

References 37 publications

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“…We show the condition α ≤ 1 − kβ is necessary. In fact, by use of the method of [26,27], we show Theorem 1.4. Suppose α > 1 − kβ, α, β ∈ R and β > 0, k is an integer and 1 ≤ k ≤ n. There exist a smooth, closed, uniformly convex hypersurface M 0 , such that under the flow (1.1),…”

Section: Introductionmentioning

confidence: 85%

A class of anisotropic expanding curvature flows

Sheng¹,

Yi²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in Euclidean R n+1 with speed u α σ β k firstly, where u is support function of the hypersurface, α, β ∈ R 1 , and β > 0, σ k is the k-th symmetric polynomial of the principle curvature radii of the hypersurface, k is an integer and 1 ≤ k ≤ n. For α ≤ 1 − kβ, β > 1 k we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for α ≤ 1 − kβ, β > 1 k , we prove that the flow with the speed f u α σ β k exists for all time and converges smoothly after normalisation to a soliton which is a solution of f u α−1 σ β k = c provided that f is a smooth positive function on S n and satisfies that (∇ i ∇ j f 2010 Mathematics Subject Classification. 35K96, 53C44.

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Section: Introductionmentioning

confidence: 85%

A class of anisotropic expanding curvature flows

Sheng¹,

Yi²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…In turn, (22) yields (23) by (21). For (24), we observe that if x ∈ Ξ K ∩ ∂ ′ K, then ν K (x), x = 0.…”

Section: On the Dual Curvature Measurementioning

confidence: 91%

The L dual Minkowski problem for p > 1 and q > 0

Böröczky

Fodor

2019

Journal of Differential Equations

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General L p dual curvature measures have recently been introduced by Lutwak, Yang and Zhang [24]. These new measures unify several other geometric measures of the Brunn-Minkowski theory and the dual Brunn-Minkowski theory. L p dual curvature measures arise from qth dual inrinsic volumes by means of Alexandrov-type variational formulas. Lutwak, Yang and Zhang [24] formulated the L p dual Minkowski problem, which concerns the characterization of L p dual curvature measures.In this paper, we solve the existence part of the L p dual Minkowski problem for p > 1 and q > 0, and we also discuss the regularity of the solution.

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“…Naturally, the dual Minkowski problem has become important for the dual Brunn-Minkowski theory introduced by Lutwak [28,29]. Since [20], progress includes a complete solution for q < 0 by Zhao [38], solutions for even µ in [4,6,15,39], and solutions via curvature flows and partial differential equations in [8,24,26].An important extension of the dual Minkowski problem was carried out by Lutwak, Yang, and Zhang [33], who introduced L p dual curvature measures and posed a corresponding L p dual Minkowski problem. In [33], the L 0 addition in [20] is replaced by L p addition, while the qth dual volume remains unchanged.…”

mentioning

confidence: 99%

General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II

et al. 2019

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The general dual volume V G (K) and the general dual Orlicz curvature measure C G,ψ (K, ·) were recently introduced for functions G : (0, ∞) × S n−1 → (0, ∞) and convex bodies K in R n containing the origin in their interiors. We extend V G (K) and C G,ψ (K, ·) to more general functions G : [0, ∞) × S n−1 → [0, ∞) and to compact convex sets K containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous dependence on the underlying set, are provided. These are required to study a Minkowski-type problem for the dual Orlicz curvature measure. We mainly focus on the case when G and ψ are both increasing, thus complementing our previous work.The Minkowski problem asks to characterize Borel measures µ on S n−1 for which there is a convex body K in R n containing the origin such that µ equals C G,ψ (K, ·), up to a constant. A major step in the analysis concerns discrete measures µ, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. For general (not necessarily discrete) measures µ, we use an approximation argument. This approach is also applied to the case where G is decreasing and ψ is increasing, and hence augments our previous work. When the measures µ are even, solutions that are originsymmetric convex bodies are also provided under some mild conditions on G and ψ. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when µ is discrete or even.2 RICHARD J. GARDNER, DANIEL HUG, SUDAN XING, AND DEPING YE problem; see e.g. [23,30,32]. The case p = 0 is of particular significance because the L 0 surface area measure, the so-called cone volume measure, is an affine invariant. The L 0 or logarithmic Minkowski problem is challenging and was only solved recently for even measures by Böröczky, Lutwak, Yang, and Zhang [5]. More recent contributions to the logarithmic Minkowski problem are [3,42] and further references and background on the L p Minkowski problem may be found in [20,36].Recent seminal work of Huang, Lutwak, Yang, and Zhang [20] brought new ingredients, the qth dual curvature measures, to the family of Minkowski problems. These measures are obtained via a first-order variation of the qth dual volume with respect to L 0 addition of convex bodies (see [20, Theorem 4.5]), the case q = n being the L 0 surface area. The authors of [20] posed a corresponding Minkowski problem-the dual Minkowski problem-of finding necessary and/or sufficient conditions for a measure µ on S n−1 to be the qth dual curvature measure of some convex body, and they provided a partial solution when µ is even. Clearly, the logarithmic Minkowski problem is a special case of the dual Minkowski problem. Naturally, the dual Minkowski problem has become important for the dual Brunn-Minkowski theory introduced by Lutwak [28,29]. Since [20], progress includes a complete solution for q < 0...

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Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Cited by 102 publications

References 37 publications

A class of anisotropic expanding curvature flows

A class of anisotropic expanding curvature flows

The L dual Minkowski problem for p > 1 and q > 0

General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II

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Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Cited by 102 publications

References 37 publications

A class of anisotropic expanding curvature flows

A class of anisotropic expanding curvature flows

The L dual Minkowski problem for p > 1 and q > 0

General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II

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The L dual Minkowski problem for p > 1 and q > 0