A note on the critical Laplace Equation and Ricci curvature

TL;DR

This paper investigates positive solutions to a critical Laplace equation on manifolds with nonnegative Ricci curvature, showing nonexistence under certain volume growth conditions unless the manifold is Euclidean space with a known solution.

Contribution

It establishes a nonexistence result for positive solutions to the critical Laplace equation on manifolds with nonnegative Ricci curvature, assuming mild volume growth conditions, and characterizes the Euclidean case.

Findings

No positive solutions exist unless the manifold is Euclidean space.

Solutions must resemble Talenti functions in the Euclidean case.

The proof uses elementary analysis of level sets of solutions.

Abstract

We study strictly positive solutions to the critical Laplace equation \[ - \Delta u = n(n-2) u^{\frac{n+2}{n-2}}, \] decaying at most like $d(o, x)^{-(n-2)/2}$, on complete noncompact manifolds $(M, g)$ with nonnegative Ricci curvature, of dimension $n \geq 3$. We prove that, under an additional mild assumption on the volume growth, such a solution does not exist, unless $(M, g)$ is isometric to $\mathbb{R}^n$ and $u$ is a Talenti function. The method employs an elementary analysis of a suitable function defined along the level sets of $u$.