Lecture notes on mean curvature flow, barriers and singular perturbations (Q2840320)

This is a beautiful elementary book on mean curvature flow and minimal barriers, containing 17 chapters and written in a very comprehensive mathematical language.NEWLINENEWLINEIn the preliminary chapter, the author, using the signed distance function \(d:\partial E\rightarrow \mathbb{R}\), for a subset \(E\) of \(\mathbb{R}^n\), introduces the mean curvature vector and the second fundamental form of \(\partial E\) and compares these definitions with the classical ones. He describes the expansion of the Hessian of the signed distance in a neighborhood of \(\partial E\), a result that will be used for proving the short-time existence and uniqueness theorem of the mean curvature flow starting from a smooth compact boundary. Then the author introduces the smooth mean curvature flows in the compact and noncompact cases, giving lots of such examples. He also gives examples of special solutions such as self-similar and translatory solutions. In order to give a variational meaning to the mean curvature flow, the author computes the first variation of some integral functionals, volume and perimeter functionals, and states the connection between the mean curvature flow and the first variation of area, being the gradient flow of the perimeter functional. Some consequences of Huisken's monotonicity formula in the study of mean curvature flow are given, namely, a maximum principle for the gradient is derived.NEWLINENEWLINEThe author presents versions of the inclusion principle between smooth mean curvature flows and characterizes a smooth mean curvature flow using the evolution of its tubular neighborhoods. The barriers and minimal barriers are introduced, motivated by the presence of singularities in the mean curvature flow. Ilmanen's interposition lemma is used to compare the weak evolution obtained using the barriers method with other notions of generalized evolution and the connection between the minimal barriers theory and the level set solutions is stated. Finally, he proves the convergence of the zero level set of solutions of the equation NEWLINE\[CARRIAGE_RETURNNEWLINE\varepsilon \frac{\partial u}{\partial t}=\varepsilon \Delta u-\frac{1}{2\varepsilon}u(u^2-1)CARRIAGE_RETURNNEWLINE\]NEWLINE to a mean curvature flow.