Noncompact $L_p$-Minkowski problems

TL;DR

This paper establishes the existence of complete, noncompact convex hypersurfaces with prescribed p-curvature functions, extending previous work for the special case p=1 to general p values.

Contribution

It provides new sufficient conditions for solving Monge-Ampère type equations related to the noncompact $L_p$-Minkowski problem for all p not equal to 1.

Findings

Existence of convex hypersurfaces with prescribed p-curvature for p ≠ 1

Extension of previous results from p=1 to general p

Sufficient conditions for solvability of related Monge-Ampère equations

Abstract

In this paper we prove the existence of complete, noncompact convex hypersurfaces whose $p$-curvature function is prescribed on a domain in the unit sphere. This problem is related to the solvability of Monge-Amp\`ere type equations subject to certain boundary conditions depending on the value of $p$. The special case of $p=1$ was previously studied by Pogorelov and Chou-Wang. Here, we give some sufficient conditions for the solvability for general $p\neq1$.