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The horospherical $p$-Christoffel-Minkowski problem in hyperbolic space

Tianci Luo, Yong Wei

TL;DR

The paper addresses the horospherical $p$-Christoffel-Minkowski problem in $\mathbb{H}^{n+1}$ by recasting it as a fully nonlinear elliptic PDE for the horospherical support function $\varphi=e^u$ and proving the existence of a smooth, uniformly $h$-convex solution under structural conditions on the prescribed function $f$. A key contribution is the full rank theorem, established via a viscosity approach inspired by $\text{Bryan–Ivaki–Scheuer}$, which ensures convexity of the solution. When $p=0$, the problem corresponds to a Nirenberg-type conformal problem on $\mathbb{S}^n$, and the results yield existence in that setting as well. The work also clarifies the connections to conformal geometry through the Schouten tensor and shows how degree theory can be employed to obtain existence, extending the Euclidean $L^p$ theory to hyperbolic space. Overall, the paper integrates horospherical geometry, nonlinear PDE methods, and conformal geometry to advance the horospherical $p$-CM problem and its geometric implications.

Abstract

The horospherical $p$-Christoffel-Minkowski problem was posed by Li and Xu (2022) as a problem prescribing the $k$-th horospherical $p$-surface area measure of $h$-convex domains in hyperbolic space $\mathbb{H}^{n+1}$. It is a natural generalization of the classical $L^p$ Christoffel-Minkowski problem in the Euclidean space $\mathbb{R}^{n+1}$. In this paper, we consider a fully nonlinear equation associated with the horospherical $p$-Christoffel-Minkowski problem. We establish the existence of a uniformly $h$-convex solution under appropriate assumptions on the prescribed function. The key to the proof is the full rank theorem, which we will demonstrate using a viscosity approach based on the idea of Bryan-Ivaki-Scheuer (2023). When $p=0$, the horospherical $p$-Christoffel-Minkowski problem in $\mathbb{H}^{n+1}$ is equivalent to a Nirenberg-type problem on $\mathbb{S}^n$ in conformal geometry. Therefore, our result implies the existence of solutions to the Nirenberg-type problem.

The horospherical $p$-Christoffel-Minkowski problem in hyperbolic space

TL;DR

The paper addresses the horospherical -Christoffel-Minkowski problem in by recasting it as a fully nonlinear elliptic PDE for the horospherical support function and proving the existence of a smooth, uniformly -convex solution under structural conditions on the prescribed function . A key contribution is the full rank theorem, established via a viscosity approach inspired by , which ensures convexity of the solution. When , the problem corresponds to a Nirenberg-type conformal problem on , and the results yield existence in that setting as well. The work also clarifies the connections to conformal geometry through the Schouten tensor and shows how degree theory can be employed to obtain existence, extending the Euclidean theory to hyperbolic space. Overall, the paper integrates horospherical geometry, nonlinear PDE methods, and conformal geometry to advance the horospherical -CM problem and its geometric implications.

Abstract

The horospherical -Christoffel-Minkowski problem was posed by Li and Xu (2022) as a problem prescribing the -th horospherical -surface area measure of -convex domains in hyperbolic space . It is a natural generalization of the classical Christoffel-Minkowski problem in the Euclidean space . In this paper, we consider a fully nonlinear equation associated with the horospherical -Christoffel-Minkowski problem. We establish the existence of a uniformly -convex solution under appropriate assumptions on the prescribed function. The key to the proof is the full rank theorem, which we will demonstrate using a viscosity approach based on the idea of Bryan-Ivaki-Scheuer (2023). When , the horospherical -Christoffel-Minkowski problem in is equivalent to a Nirenberg-type problem on in conformal geometry. Therefore, our result implies the existence of solutions to the Nirenberg-type problem.

Paper Structure

This paper contains 8 sections, 15 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The elementary symmetric functions
  4. Horospherically convex domains and hypersurfaces
  5. Relation to the conformal metric on $\mathbb{S}^n$
  6. A priori estimates
  7. Full rank theorem
  8. Proof of Theorem \ref{['Main']}

Key Result

Theorem 1.4

Let $n\ge2$ and $1\le k\le n-1$ be integers. Let $p\geq 0$ be a real number. Given a smooth positive even function $f(x)$ on $\mathbb{S}^n$. If $f(x)$ satisfies the Assumption assum, then equation equa has a smooth, even and uniformly h-convex solution $\varphi>1$.

Theorems & Definitions (26)

  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.6
  • Lemma 3.1
  • ...and 16 more