# Conformally Warped Manifolds and quasi-Einstein metrics

# Conformally Warped Manifolds and quasi-Einstein metrics

The concept of a smooth metric measure space has recently arisen as a useful object within Riemannian geometry, for example in Perelman's formulation of Ricci flow as a gradient flow. In this setting, a key objective is to find a suitable generalization of Ricci curvature, and to understand the associated ``quasi-Einstein'' metrics. Taking two different perspectives, Lott, Villani, Sturm and Chang, Gursky and Yang have found two distinct approaches to studying smooth metric measure spaces. While the formulations are different, they both introduce an extra dimensional parameter $m$ which, in the limit $m\to\infty$, recovers the curvatures that arise in Perelman's treatment of the Ricci flow. In this way it becomes interesting to see if the two approaches are related. As the quasi-Einstein metrics of these approaches include conformally Einstein metrics, the bases of Einstein warped products, and gradient Ricci solitons, finding a relation between them might also allow us to find interesting connections between these metrics.In this talk, I will introduce what I call ``conformally warped manifolds'' as a way to unite the approaches of Lott-Villani-Sturm and Chang-Gursky-Yang. I will discuss three results which suggest that this notion is indeed the ``best'' approach. First, I will discuss the variational problem associated to quasi-Einstein metrics, which naturally relates the Yamabe constant to Perelman's shrinker entropy. Second, I will discuss a Liouville-type theorem which illustrates the usefulness of studying the limit $m\to\infty$ as a way to overcome difficulties in the $m=\infty$ comparison theory. Third, I will discuss a compactness theorem for compact quasi-Einstein metrics analogous to Anderson's theorem for Einstein metrics, which is independent of the parameter $m$.