The horospherical $p$-Christoffel-Minkowski and prescribed $p$-shifted Weingarten curvature problems in hyperbolic space

TL;DR

This paper extends key curvature problems from Euclidean to hyperbolic space, proving existence of smooth convex solutions for the horospherical $p$-Christoffel-Minkowski and prescribed $p$-shifted Weingarten curvature problems.

Contribution

It introduces and solves the hyperbolic space versions of the $L_p$-Christoffel-Minkowski and prescribed $L_p$-Weingarten curvature problems, establishing new existence theorems.

Findings

Existence of smooth, origin-symmetric, strictly horospherically convex solutions.

Development of a new full rank theorem for hyperbolic space.

Proven existence results for the prescribed $p$-shifted Weingarten curvature problem.

Abstract

The $L_p$-Christoffel-Minkowski problem and the prescribed $L_p$-Weingarten curvature problem for convex hypersurfaces in Euclidean space are important problems in geometric analysis. In this paper, we consider their counterparts in hyperbolic space. For the horospherical $p$-Christoffel-Minkowski problem first introduced and studied by the second and third authors, we prove the existence of smooth, origin-symmetric, strictly horospherically convex solutions by establishing a new full rank theorem. We also propose the prescribed $p$-shifted Weingarten curvature problem and prove an existence result.