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Geometric flows have many applications in physics and geometry. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds.
Formation of Singularities in the Mean Curvature Flow.- Geometry of hypersurfaces.- Examples.- Local existence and formation of singularities.- Invariance properties.- Singular behaviour of convex surfaces.- Convexity estimates.- Rescaling near a singularity.- Huisken’s monotonicity formula.- Cylindrical and gradient estimates.- Mean curvature flow with surgeries.- Geometric Flows, Isoperimetric Inequalities and Hyperbolic Geometry.- The classical isoperimetric inequality in Euclidean space.- Surfaces.- Higher dimensions.- Some applications to hyperbolic geometry.