Lecture notes on mean curvature flow, barriers and singular perturbations

The aim of this book is to give an introduction to mean curvature flow using, as much as possible, a parametrization free approach. Some relevant aspects of mean curvature flow are described, such as the role of the signed distance function and the comparison principle, and their use in the theory of barriers. Some simple examples of singularities are discussed. Finally, also making use of a formal asymptotic inner and outer expansion, we prove the convergence of the parabolic Allen-Cahn's equation to mean curvature flow for sufficiently short times, together with an error estimate