Non-normalized solutions to the horospherical Minkowski problem

TL;DR

This paper addresses the horospherical $p$-Minkowski problem in hyperbolic space, establishing existence results for solutions without normalization for a range of p values using degree theory.

Contribution

It introduces a degree-theoretic approach to remove the normalization factor in solving the horospherical $p$-Minkowski problem for certain p values, advancing the understanding of this geometric problem.

Findings

Existence of solutions for all p in [-n, n) without normalization.

Development of a degree-theoretic method for the problem.

Overcoming the challenge posed by the lack of homogeneity.

Abstract

Recently, the horospherical $p$-Minkowski problem in hyperbolic space was proposed as a counterpart of $L_p$ Minkowski problem in Euclidean space. Through designing a new volume preserving curvature flow, the existence of normalized even solution to the horospherical $p$-Minkowski problem was solved for all $p \in \mathbb{R}$. However, due to the lack of homogeneity of the horospherical $p$-surface area measure, it is difficult to remove the normalizing factor. In this paper, we overcome this difficulty for $-n\leq p<n$ by the degree-theoretic approach.