Lambda constant and Ground states of Perelman's W-functional

TL;DR

This paper investigates the generalized lambda constant and the existence of ground states for Perelman's W-functional on non-compact Riemannian manifolds, using variational methods and concentration-compactness techniques.

Contribution

It provides new estimates for the lambda constant and establishes the existence of ground states for generalized functionals on certain non-compact manifolds.

Findings

Estimation of the generalized lambda constant.

Existence of ground states for generalized F- and W-functionals.

Application of Lions' concentration-compactness method.

Abstract

In this paper, we consider the generalized lambda constant and the existence of ground states of the generalized Perelman's W-functional from a variational formulation. One result is concerned with the estimation of the generalized $\lambda$ constant. The other results are about the existence of ground states of generalized $F$-functional and W-functional both on a complete non-compact Riemannian manifold $(M,g)$ with positive injectivity radius and with Ricci curvature bounded from below. Our main results are Theorems 2,3 and 7. For the existence of the ground states we use Lions' concentration-compactness method.