New Results About the Lambda Constant and Ground States of the 𝑊-Functional

Abstract

Abstract In this paper, we study properties of the lambda constants and the existence of ground states of Perelman’s famous W-functional from a variational formulation. We have two kinds of results. One is about the estimation of the lambda constant of G. Perelman, and the other is about the existence of ground states of his W-functional, both on a complete non-compact Riemannian manifold ( M , g ) {(M,g)} . One consequence of our estimation is that, on an ALE (or asymptotic flat) manifold ( M , g ) {(M,g)} , if the scalar curvature s of ( M , g ) {(M,g)} is non-negative and has quadratical decay at infinity, then M is scalar flat, i.e., s = 0 {s=0} in M. We also introduce a new constant d ⁢ ( M , g ) {d(M,g)} . For the existence of the ground states, we use Lions’ concentration-compactness method.