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 PoincarÃ© 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 KÃ¤hler-Ricci flow, and Yau's uniformization conjecture.  Contents  On the conformal scalar curvature equation and related problems   A survey of the KÃ¤hler-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 TeichmÃ¼ller 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

Surveys in Differential Geometry, Vol. 12: Geometric Flows (International Press) PDF

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