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

Huan-Jie Chen, Shi-Zhong Du

TL;DR

The paper addresses the symmetry classification of the $L_p$-Minkowski problem on the sphere by projecting the spherical equation to a Euclidean Monge–Ampère form and applying Lie group analysis to determine all invariant transformations. The main approach combines a semi-spherical projection to reduce to $\det D^2u=(1+|x|^2)^{-(p+n+1)/2}u^{p-1}$, a thorough Lie symmetry study to extract infinitesimal generators, and a reconstruction of corresponding convex-body transformations. The key contributions are an explicit, complete classification of the symmetry group across all real $p$: rotations and scalings for $p=n+1$, rotations and translations for $p=1$, centro-affine for $p=-n-1$, and rotations for other $p$, together with a global–local compatibility framework. This symmetry atlas enhances understanding of the geometric structure of the $L_p$-Minkowski problem and informs symmetry-based solution strategies in convex geometry.

Abstract

In Convex Geometry, a core topic is the $L_p$-Minkowski problem \begin{equation}\label{e0.1} \det(\nabla^2h+hI)=fh^{p-1}, \ \ \forall X\in{\mathbb{S}}^n, \ \ \forall p\in \mathbb{R} \end{equation} of Monge-Ampère type. By the transformation $u(x)=h(X)\sqrt{1+|x|^2}$ and semi-spherical projection, equation \eqref{e0.1} can be reformulated by the Monge-Ampère type equation \begin{equation}\label{e0.2} \det D^2u=(1+|x|^2)^{-\frac{p+n+1}{2}}u^{p-1}, \ \ \forall x\in{\mathbb{R}}^n, \ \ \forall p\in \mathbb{R} \end{equation} on the Euclidean space. In this paper, we will firstly determine the symmetric groups of $n$-dimensional fully nonlinear equation \eqref{e0.2} 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.

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

TL;DR

The paper addresses the symmetry classification of the -Minkowski problem on the sphere by projecting the spherical equation to a Euclidean Monge–Ampère form and applying Lie group analysis to determine all invariant transformations. The main approach combines a semi-spherical projection to reduce to , a thorough Lie symmetry study to extract infinitesimal generators, and a reconstruction of corresponding convex-body transformations. The key contributions are an explicit, complete classification of the symmetry group across all real : rotations and scalings for , rotations and translations for , centro-affine for , and rotations for other , together with a global–local compatibility framework. This symmetry atlas enhances understanding of the geometric structure of the -Minkowski problem and informs symmetry-based solution strategies in convex geometry.

Abstract

In Convex Geometry, a core topic is the -Minkowski problem \begin{equation}\label{e0.1} \det(\nabla^2h+hI)=fh^{p-1}, \ \ \forall X\in{\mathbb{S}}^n, \ \ \forall p\in \mathbb{R} \end{equation} of Monge-Ampère type. By the transformation and semi-spherical projection, equation \eqref{e0.1} can be reformulated by the Monge-Ampère type equation \begin{equation}\label{e0.2} \det D^2u=(1+|x|^2)^{-\frac{p+n+1}{2}}u^{p-1}, \ \ \forall x\in{\mathbb{R}}^n, \ \ \forall p\in \mathbb{R} \end{equation} on the Euclidean space. In this paper, we will firstly determine the symmetric groups of -dimensional fully nonlinear equation \eqref{e0.2} without asymptotic growth assumption. After proving several key resolution lemmas, we thus completely classify the symmetric groups of the -Minkowski problem. Our method develops the Lie theory to fully nonlinear PDEs in Convex Geometry.

Paper Structure

This paper contains 7 sections, 14 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminary to Lie's theory on partial differential equations
  3. Projecting of the $L_p$-Minkowski problem from ${\mathbb{S}}^n_-$ to ${\mathbb{R}}^n$
  4. Resolution transformations on convex body for $p=n+1$
  5. Resolution transformations on convex body for $p=1$
  6. Resolution transformations on convex body for $p=-n-1$
  7. Symmetry group of the $L_p$-Minkowski problem

Key Result

Theorem 1.1

Considering the $L_p$-Minkowski problem e1.2, one has (1) when $p>n+1$, there exists a unique positive solution in $C^{2,\alpha}({\mathbb{S}}^n)$ for each positive function $f\in C^\alpha({\mathbb{S}}^n), \alpha\in(0,1)$; (2) when $p=n+1$, there exists a unique pair $(h,\lambda)$ for $0<h\in C^{2+\a for each positive function $f\in C^\alpha({\mathbb{S}}^n), \alpha\in(0,1)$; (3) when $1<p<n+1$, e1.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 6.1
  • ...and 5 more