Complete Classification of the Symmetry Group of L_p-Minkowski Problem on the Sphere

2025/04/03 by Chen, Huan-Jie, Du, Shi-Zhong
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.02661

Abstract

In Convex Geometry, a core topic is the L_p-Minkowski problem det(∇²h+hI)=fh^p-1, ∀ X∈\mathbbSⁿ, ∀ p∈ ℝ of Monge-Ampère type. By the transformation u(x)=h(X)√(1+|x|²) and semi-spherical projection, equation \eqrefe0.1 can be reformulated by the Monge-Ampère type equation det D²u=(1+|x|²)^-(p+n+1)/(2)u^p-1, ∀ x∈ℝⁿ, ∀ p∈ ℝ on the Euclidean space. In this paper, we will firstly determine the symmetric groups of n-dimensional fully nonlinear equation \eqrefe0.2 without asymptotic growth assumption. After proving several key resolution lemmas, we thus completely classify the symmetric groups of the L_p-Minkowski problem. Our method develops the Lie theory to fully nonlinear PDEs in Convex Geometry.

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