The energy of a smooth metric measure space and applications

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.