Lichnerowicz-type equations on complete manifolds

TL;DR

This paper establishes existence, bounds, and uniqueness results for positive solutions of Lichnerowicz-type equations on complete manifolds without relying on curvature assumptions.

Contribution

It provides new existence and uniqueness theorems for Lichnerowicz-type equations under spectral conditions, without curvature restrictions.

Findings

Proved existence of positive solutions under spectral assumptions

Established a priori bounds for solutions

Demonstrated uniqueness in certain cases

Abstract

Under appropriate spectral assumptions we prove two existence results for positive solutions of Lichnerowicz-type equations on complete manifolds. We also give a priori bounds and a comparison result that immediately yields uniqueness for certain classes of solutions. No curvature assumptions are involved in our analysis.