Geometric evolution equations

In this lecture series we present a number of evolution equations arising in geometry (mean curvature flow, Willmore flow, Yamabe flow, harmonic map flow). We will first state and discuss basic knowledge from differential geometry and from the theory of evolving surfaces. Then selfsimilar solutions and qualitative properties of the flows will be discussed (formation of singularities, comparison principles, gradient flow properties). We will then study existence and regularity theory taking Campanato theory into account. Furthermore we discuss the evolution of geometric quantities in order to analyze the structure of singularities for mean curvature flow. Finally we prove Huisken's monotonicity formula and state some consequences.