Introduction to Mean Curvature Flow

Yoshihiro Tonegawa**, Alessandra Pluda

**Yoshihiro Tonegawa is Professor at the Department of Mathematics, Tokyo Institute of Technology.

Abstact: A family of surfaces is called the Mean Curvature Flow (MCF) if the velocity of a surface is equal to the mean curvature at each point and time. It is one of the most important geometric evolution problems with many facets of studies such as analysis of singularities, notions of weak solution, solvability of initial value problems, and so forth, and the course touches upon some of the recent developments.
The course consists of two parts. The first part is on the MCF in the framework of Geometric Measure Theory called the Brakke flow. Starting from the definitions and preliminaries, many of the basic properties of Brakke flow as well as some advanced topics such as the general existence theory will be covered.
The second part focuses on classical results on MCF obtained with a PDE approach. The main topics are short-time existence, the maximum principle, evolution equations of geometric quantities, and the analysis of type I and II singularities in the special case of positive mean curvature. In particular, the case of planar curves will be analyzed in full detail.
Some familiarity with measure theory and parabolic equations is desirable but not necessary.

Main references:
*Tonegewa, Yoshihiro, Brakke’s Mean Curvature Flow: An Introduction, Springer Briefs in Mathematics, Springer, 2019
*Mantegazza, Carlo, Lecture notes on mean curvature flow, Progress in Mathematics, Birkhäuser/Springer Basel AG, Basel, 2011.

Lectures:

14/11/2022: Introduction to Brakke flow, preliminaries from Geometric Measure Theroy (YOSHIHIRO TONEGAWA)
15/11/2022: Huisken's monotonicity formula and its applications (YOSHIHIRO TONEGAWA)
16/11/2022: Tangent flows and regularity theory. (YOSHIHIRO TONEGAWA)
22/11/2022: Existence theorem of Brakke flow (YOSHIHIRO TONEGAWA)
23/11/2022: Construction of approximate flow (YOSHIHIRO TONEGAWA)
25/11/2022: Compactness theorem of time discrete approximate flow (YOSHIHIRO TONEGAWA)
29/11/2022: First variation of the area functional. The mean curvature flow can be understood as the geometric gradient flow of the area functional. Derivation of the equations of mean curvature flow and of the PDE system of the network flow. (ALESSANDRA PLUDA)
01/12/2022: Invariance under diffeomorphisms of the mean curvature flow. Invariance under reparametrization of the network flow, choice of the tangential component of the velocity. List of possible statements of well-posedness theorems of the network flow in Sobolev and Hölder spaces. Compatibility conditions for the network flow. (ALESSANDRA PLUDA)
06/12/2022: Examples of solutions to the Mean Curvature Flow: sphers, cilinders, minimal surfaces, translators, shrinkers e expanders. A notion of dynamical instability for the Mean Curvature Flow. Mean Curvature Flow of entire graphs. Evolution of the area enlcosed in a loop of a network. Examples of singularity of the network flow: i) in finite time the area enclosed in a loop vanishes, the length of the curves of the loop goes to zero and the L^2 norm of the curvature blows up. ii) Possible behaviors of a symmetric networks composed of five curves. (ALESSANDRA PLUDA)
15/12/2022: Short time existence for the Mean Curvature Flow. Reduction to a scalar equation for the height function, linearization and fixed point arguments to solve a quasilinear equation. (ALESSANDRA PLUDA)
19/12/2022: Solution of the heat equation with Dirichlet boundary condition in weigthed Sobolev spaces with Lax-Milgram Lemma, a priori estimates and estimates on the difference quotiences, regularity. (ALESSANDRA PLUDA)
24/01/2023: Comparison principle e consequences. Evolution of geometric quantities. (ALESSANDRA PLUDA)
26/01/2023: Integral estimates for derivaties of the curvature in the network flow. (ALESSANDRA PLUDA)
31/01/2023: Proof of Grayson's Theorem. (ALESSANDRA PLUDA)
14/02/2023: Restarting the network flow after a singularity: parabolic blow-up, change of coordinates, the lifted equations and identification of a function from the Taylor coefficients of the boundary componets with Borel's Lemma. (ALESSANDRA PLUDA)

Seminars:

Cristian Sopio

*Ecker, Klaus; Huisken, Gerhard, "Mean curvature evolution of entire graphs". Ann. of Math. (2) 130 (1989), no. 3, 453–471.

Lorenzo Ferreri

*Kim, Lami; Tonegawa, Yoshihiro, "Existence and regularity theorems of one-dimensional Brakke flows." Interfaces Free Bound. 22 (2020), no. 4, 505–550.

Giorgia Benassi

*Huisken, Gerhard. "A distance comparison principle for evolving curves." Asian J. Math. 2 (1998), no. 1, 127–133.

*Mantegazza, Carlo, Lecture notes on mean curvature flow, Progress in Mathematics, Birkhäuser/Springer Basel AG, Basel, 2011 - part of chapter 3: "Monotonicity Formula and Type I Singularities".

Luciano Sciaraffia

* Lira, Jorge, Mazzeo, Rafe, Pluda, Alessandra and Sáez, Mariel , "Short-time existence for the network flow", Comm. Pure Appl. Math. 76 (2023), no. 12, 3968–4021.

Roberto Colombo

*Andrews, Ben and Bryan, Paul "Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem" J. Reine Angew. Math. 653 (2011), 179–187.