The energy of a smooth metric measure space and applications

TL;DR

This paper introduces a new energy concept for smooth metric measure spaces, linking it to known invariants like the Yamabe constant and Perelman's entropy, and applies it to prove a precompactness theorem for quasi-Einstein spaces.

Contribution

It defines the energy of smooth metric measure spaces, explores its properties, and uses it to establish a precompactness theorem for quasi-Einstein spaces.

Findings

Energy relates to Yamabe constant and Perelman's entropy

Energy shares properties with these invariants

Precompactness theorem for quasi-Einstein spaces proved

Abstract

We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's $\nu$-entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the $\kappa$-noncollapsing property. Finally, we use the energy to prove a precompactness theorem for the space of compact quasi-Einstein smooth metric measure spaces, in the spirit of similar results for Einstein metrics and gradient Ricci solitons.