InhaltsangabeThe classical isoperimetric inequality in Euclidean space. Three different approaches.- The curve shortening flow and isoperimetric inequalities on surfaces.- $H^k$-flows and isoperimetric inequalities.- Estimates on the Willmore functional and isoperimetric inequalities.- Singularities in the volume-preserving mean curvature flow.- Bounds on the Heegaard genus of a hyperbolic manifold.- The isoperimetric profile for small volumes.- Local existence of flows driven by the second fundamental form and formation of singularities.- Invariance properties.- Singular behaviour of convex surfaces.- Convexity estimates.- Rescaling near a singularity.- Cylindrical and gradient estimates.- Mean curvature flow with surgeries.