Surveys in Differential Geometry (Volume 12): Geometric Flows

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. Contents On the conformal scalar curvature equation and related problems A survey of the Kähler-Ricci Flow and Yau's Uniformization Conjecture Recent developments on the Hamilton's Ricci Flow Curvature flows in semi-Riemannian manifolds Global regularity and singularity development for wave maps Relativistic Teichmüller theory: a Hamilton-Jacobi approach to 2+1 dimensional Einstein gravity Monotonicity and Li-Yau-Hamilton inequalities Singularities of mean curvature flow and flow with surgeries Some recent developments in Lagrangian mean curvature flows