Paper · Dec 18, 2024

The $L_{p}$ Gauss dual Minkowski problem

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Abstract

This article introduces the $L_{p}$ -Gauss dual curvature measure and proposes its related $L_{p}$ -Gauss dual Minkowski problem as: for $p, q \in R$ , under what necessary and/or sufficient condition on a non-zero finite Borel measure $μ$ on unit sphere does there exist a convex body $K$ such that $μ$ is the $L_{p}$ Gauss dual curvature measure? If $K$ exists, to what extent is it unique? This problem amounts to solving a class of Monge-Ampère type equations on unit sphere in smooth case: \beginalign e^{-\frac{|\nabla h_K|^2+h_K^2}2}h_K^1-p (|\nabla h_K|^2+h_K^2)^{\fracq-n2} \det(\nabla^2h_K+h_KI)=f,\qquad (0.1) \endalign where $f$ is a given positive smooth function on unit sphere, $h_{k}$ is the support function of convex body $K$ , $\nabla h_{K}$ and $\nabla^{2} h_{K}$ are the gradient and Hessian of $h_{K}$ on unit sphere with respect to an orthonormal basis, and $I$ is the identity matrix. We confirm the existence of solution to the new problem with $p, q > 0$ and the existence of smooth solution to the equation (0.1) with $p, q \in R$ by variational method and Gaussian curvature flow method, respectively. Furthermore, the uniqueness of solution to the equation (0.1) in the case $p, q \in R$ with $q < p$ is established.

Comments: 25 pages

Other curvature flows Gauss curvature flow Kähler geometry Stability & uniqueness

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The $L_{p}$ Gauss dual Minkowski problem

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The Lp​ Gauss dual Minkowski problem

The $L_{p}$ Gauss dual Minkowski problem