Uniqueness of Positive Solutions of the Conformal Scalar Curvature Equation and Applications to Conformal Transformations

TL;DR

This paper investigates the uniqueness of positive solutions to the conformal scalar curvature equation on negatively curved manifolds and explores implications for conformal transformations and solution types.

Contribution

It establishes conditions for uniqueness of solutions and demonstrates that certain conformal transformations are actually isometries on these manifolds.

Findings

Uniqueness of positive solutions under specific conditions

Conformal transformations are isometries in certain cases

Results on radial and complete solutions

Abstract

We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We also study uniqueness of complete positive solutions and radial solutions.