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Flow by Gauss curvature to the $L_p$-Gaussian Minkowski problem

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arxiv 2212.01822 v1 pith:UCGQMQI2 submitted 2022-12-04 math.DG math.AP

Weimin Sheng , Ke Xue This is my paper · ORCID
classification math.DGmath.AP
keywords gaussianminkowskiproblemgausscasescurvatureevenexistence
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abstract

In this paper, we study the $L_p$-Gaussian Minkowski problem, which arises in the $L_p$-Brunn-Minkowski theory in Gaussian probability space. We use Aleksandrov's variational method with Lagrange multipliers to prove the existence of the logarithmic Gauss Minkowski problem. We construct a suitable Gauss curvature flow of closed, convex hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$, and prove its long-time existence and converges smoothly to a smooth solution of the normalized $L_p$ Gaussian Minkowski problem in cases of $p>0$ and $-n-1<p\leq 0$ with even prescribed function respectively. We also provide a parabolic proof in the smooth category to the $L_p$-Gaussian Minkowski problem in cases of $p\geq n+1$ and $0<p<n+1$ with even prescribed function, respectively.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Gaussian Minkowski problem for epigraphs of convex functions

    math.FA 2025-08 unverdicted novelty 6.0 of 10

    The Gaussian Minkowski problem is generalized to epigraphs of convex functions, and existence of convex functions realizing prescribed Gaussian moment measures is established under mild conditions.

  2. Uniqueness of solutions to the isotropic $L_{p}$ Gaussian Minkowski problem

    math.AP 2024-12 accept novelty 6.0 of 10

    For -(n+1)<p<-1, isotropic Lp Gaussian Minkowski solutions with R(K)≤1 are unique and spherical, without assuming the body is origin-centered.

  3. The even Lp Gaussian dual Minkowski problem

    math.FA 2024-12 reject novelty 4.0 of 10

    For p,q>1, an origin-symmetric convex body is claimed to exist whose normalized Lp Gaussian dual curvature measure matches any given even density measure, up to total mass scaling.

  4. Minkowski Problems for Geometric Measures

    math.MG 2025-02 unverdicted novelty 2.0 of 10

    A comprehensive survey that organizes the Minkowski problems of convex geometry into a unified framework based on geometric measures as differentials of global geometric invariants.

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