Extrinsic geometric flows are characterized by a submanifold evolving in an ambient space with velocity determined by its extrinsic curvature. The goal of this book is to give an extensive introduction to a few of the most prominent extrinsic flows, namely, the curve shortening flow, the mean curvature flow, the Gauss curvature flow, the inverse-mean curvature flow, and fully nonlinear flows of mean curvature and inverse-mean curvature type. The authors highlight techniques and behaviors that frequently arise in the study of these (and other) flows. To illustrate the broad applicability of...

Ricci Solitons in Dimensions $4$ and Higher offers a detailed account of recent developments of Ricci solitons-self-similar solutions to the Ricci flow equation-which play a central role in modeling the formation of singularities of the flow. Building on the foundational work of Hamilton and Perelman and the recent advances of Bamler, Brendle, and others, this book focuses on the rich and technically demanding theory of these solutions. With special attention to dimension $4$-where potential applications to the topology of smooth 4-manifolds are most promising-the authors present key...