Geometric flows are non-linear parabolic differential equations which describe the evolution of geometric structures. Inspired by Hamilton’s Ricci flow, the field of geometric flows has seen tremendous progress in the past 25 years and yields important applications to geometry, topology, physics, nonlinear analysis, and so on. Of course, the most spectacular development is Hamilton’s theory of Ricci flow and its application to three-manifold topology, including the Hamilton-Perelman proof of the Poincaré conjecture.

This twelfth volume of the annual Surveys in Differential Geometry examines recent developments on a number of geometric flows and related subjects, such as Hamilton’s Ricci flow, formation of singularities in the mean curvature flow, the Kähler-Ricci flow, and Yau’s uniformization conjecture.