A note on the critical Laplace Equation and Ricci curvature

Abstract

We study strictly positive solutions to the critical Laplace equationon complete noncompact manifolds (M, g) with nonnegative Ricci curvature, of dimension n ≥ 3. We prove that, under an additional mild assumption on the volume growth, such a solution does not exist, unless (M, g) is isometric to R n and u is a Talenti function. The method employs an elementary analysis of a suitable function defined along the level sets of u.