The Yamabe equation on complete manifolds with finite volume

TL;DR

This paper establishes the existence of solutions to the Yamabe equation on complete, finite-volume manifolds with positive Yamabe invariant, using approximation by eigenfunctions, and offers a new proof for the closed manifold case.

Contribution

It introduces a novel approximation method for solving the Yamabe equation on non-compact manifolds, extending known results and providing a new proof for the compact case.

Findings

Existence of Yamabe solutions on complete finite-volume manifolds

New approximation technique using eigenfunctions

Alternative proof for closed manifolds with positive Yamabe invariant

Abstract

We prove the existence of a solution of the Yamabe equation on complete manifolds with finite volume and positive Yamabe invariant. In order to circumvent the standard methods on closed manifolds which heavily rely on global (compact) Sobolev embeddings we approximate the solution by eigenfunctions of certain conformal complete metrics. This also gives rise to a new proof of the well-known result for closed manifolds and positive Yamabe invariant.