Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations

The book "Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations" by Giovanni Bellettini offers a comprehensive examination of geometric developments, particularly the Mean Curvature Flow and anisotropic Mean Curvature Flow of hypersurfaces. It analyzes the formation of these flows as well as their geometric and variational properties. Key topics include the comparison principle and its application to the definition of suitable weak solutions. Additionally, the anisotropic evolution is considered from the perspective of Finsler geometry. Another important aspect of the book is the discussion of singular perturbations, particularly the convergence of Allen-Cahn (or Ginsburg-Landau) type equations to (possibly anisotropic) Mean Curvature Flows before singularities arise in the boundary problem. The work also addresses asymptotic issues in the static case and demonstrates convergence to specified curvature problems.