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Uniqueness in the near isotropic Lp dual Minkowski problem

Karoly J. Boroczky, Shibing Chen, Weiru Liu, Christos Saroglou

TL;DR

The paper advances the isotropic understanding of the $L_p$-$q$ dual Minkowski problem on $S^{n-1}$ by proving local uniqueness when the dual curvature is close to constant and $q$ is near $n$, for $p$ in $(-1,1)$. It develops sharp $C^0$ estimates that are optimal for $p<-1$ and $q=n$, and extends near-isotropic uniqueness to the $o$-symmetric setting for $-1<p<q< ext{min}igig olinebreakig(n,n+pig)$ and $q>0$. The approach combines regularity theory for Monge-Ampère equations on the sphere, compactness arguments, and an implicit-function-type argument around the isotropic solution, along with precise control of dual curvature measures under linear transforms. These results broaden the scope of uniqueness in dual Minkowski problems, offering stability insights and new techniques for handling near-constant data in a nontrivial geometric-analytic setting.

Abstract

For n>1 and -1<p<1, we prove that if q is close to n and the qth Lp dual curvature is Holder close to be the constant one function, then this "near isotropic" qth Lp dual Minkowski problem on the (n-1)-dimensional sphere has a unique solution. Along the way, we establish a C0 estimate for -1<p<1 that is optimal in the sense that if p<-1 and q=n, then it is known that the analogous C0 estimate does not hold. We also prove the uniqueness of the solution of the near isotropic even qth Lp dual Minkowski problem on the (n-1)-dimensional sphere if -1<p<q<min{n,n+p} and q>0.

Uniqueness in the near isotropic Lp dual Minkowski problem

TL;DR

The paper advances the isotropic understanding of the - dual Minkowski problem on by proving local uniqueness when the dual curvature is close to constant and is near , for in . It develops sharp estimates that are optimal for and , and extends near-isotropic uniqueness to the -symmetric setting for and . The approach combines regularity theory for Monge-Ampère equations on the sphere, compactness arguments, and an implicit-function-type argument around the isotropic solution, along with precise control of dual curvature measures under linear transforms. These results broaden the scope of uniqueness in dual Minkowski problems, offering stability insights and new techniques for handling near-constant data in a nontrivial geometric-analytic setting.

Abstract

For n>1 and -1<p<1, we prove that if q is close to n and the qth Lp dual curvature is Holder close to be the constant one function, then this "near isotropic" qth Lp dual Minkowski problem on the (n-1)-dimensional sphere has a unique solution. Along the way, we establish a C0 estimate for -1<p<1 that is optimal in the sense that if p<-1 and q=n, then it is known that the analogous C0 estimate does not hold. We also prove the uniqueness of the solution of the near isotropic even qth Lp dual Minkowski problem on the (n-1)-dimensional sphere if -1<p<q<min{n,n+p} and q>0.
Paper Structure (7 sections, 16 theorems, 184 equations)

This paper contains 7 sections, 16 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Some basic notation and notions
  3. Regularity if the dual Minkowski curvature is bounded and bounded away from zero
  4. Proof of Theorem \ref{['qcloseton-dualcurvcloseto1']}
  5. The basic estimate for $o$-symmetric convex bodies when $-1<p<q<\min\{n,n+p\}$ and $q>0$
  6. Proof of Theorem \ref{['lambda-Cpq-symmetric']}
  7. Proof of Theorems \ref{['local-uniqueness-Cpq']} and \ref{['local-uniqueness-even-Cpq']}

Key Result

THEOREM 1.1

For $n\geq 2$, $\alpha\in(0,1)$ and $p\in[0,1)$, there exists a constant $\varepsilon_0>0$, that depends only on $n$, $\alpha$, $p$, such that if $f\in C^{\alpha}(S^{n-1})$ satisfies $\|f-1\|_{C^{\alpha}(S^{n-1})}<\varepsilon_0$, then the equation has a unique solution in ${\cal K}^n_o$ in the sense of Alexandrov, and $u=h_K|_{S^{n-1}}$ is a positive $C^{2,\alpha}$ solution in LpMonge.

Theorems & Definitions (22)

  • THEOREM 1.1: BoS24, CFL22, HuI25
  • THEOREM 1.2: Huang, Zhao
  • THEOREM 1.3
  • THEOREM 1.5
  • THEOREM 1.6
  • THEOREM 1.7
  • REMARK 2.1
  • THEOREM 3.1: Caffarelli
  • LEMMA 3.2
  • CLAIM 4.1
  • ...and 12 more