**Instructor:** Prof. Robert (Bob) Haslhofer

**Contact Information:** roberth(at)math(dot)toronto(dot)edu, BA6208

**Lectures:** Monday 13--15 and Wednesday 11--12 at the Fields Institute (Stewart Library)

**Office Hours:** Monday 10--12 in BA6208 (or by appointment)

**Website:** http://www.math.toronto.edu/roberth/mcf.html

**Course Description:** A surface moving so as to decrease its area most efficiently is said to evolve by mean curvature flow. Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares many features with Hamilton's Ricci flow from intrinsic geometry. Mean curvature flow and its variants have some striking applications in geometry, topology, material science, image processing and general relativity.

In this course we will provide a general introduction to mean curvature flow and its applications. In the first half of the course we will focus on the evolution up to the first singular time, which can be described in the classical framework of smooth differential geometry and partial differential equations. A central challenge is then to extend the solutions beyond the first singular time and to analyze the structure of singularities. This is crucial for the most striking applications, and will be our focus for the second half of the course.

**Prerequisites:** Some basic background in geometry is required, i.e. you should be familiar with basic concepts such as surfaces, manifolds, Riemannian metrics and the second fundamental form. Some prior knowledge of PDEs is also helpful, but not strictly required.

**Grading Scheme:** Attendance and participation 20%, Assignments 30%, Exam 50%

**References:**

K. Ecker: *Regularity Theory for Mean Curvature Flow*, Birkhauser, 2004

T. Ilmanen: *Elliptic regularization and partial regularity for motion by mean curvature*, Memoirs of the AMS, 1994

C. Mantegazza: *Lecture Notes on Mean Curvature Flow*, Birkhauser, 2012

B. White: *Lecture notes on Mean Curvature flow* (lecture notes by O. Chodosh)

B. Andrews: Noncollapsing in mean-convex mean curvature flow, Geom. Topol. 16(3):1413--1418, 2012.

L. Evans, J. Spruck: Motion of level sets by mean curvature. I., J. Differential Geom. 33(3):635--681, 1991.

R. Hamilton: Convex hypersurfaces with pinched second fundamental form, Comm. Anal. Geom. 2(1):167--172, 1994.

R. Haslhofer, B. Kleiner: *Mean curvature flow of mean convex hypersurfaces*, Comm. Pure Appl. Math. 70(3):511--546, 2017.

G. Huisken: Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20(1):237--266, 1984.

G. Huisken: Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31(1):285--299, 1990.

G. Huisken, C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. PDE 8(1):1--14, 1999.

B. White: Partial regularity of mean-convex hypersurfaces flowing by mean curvature, Internat. Math. Res. Notices 1994(4):185--192, 1994.

B. White: A local regularity theorem for mean curvature flow, Ann. of Math. 161(3):1487--1519, 2005.

**WEEKLY SCHEDULE**

**Week 1 (Sep 18-24)**

Mo: Introduction and motivation - Curve shortening flow, Mean curvature flow

We: Review - Geometry of hypersurfaces

**Week 2 (Sep 25-Oct 1)**

Mo: Evolution equations under MCF, maximum principle

We: short time existence

We 2pm: *GA colloquium talk on a recent application of MCF*

**Week 3 (Oct 2-Oct 8)**

Mo: Derivative estimates, maximal existence time

We: local derivative estimates

**Week 4 (Oct 9-Oct 15)**

Mo: no class (Thanksgiving)

We: Preserved curvature conditions

**Week 5 (Oct 16-Oct 22)**

Mo: Huisken's monotonicity formula, type I blowups

We: Classification of mean convex shrinkers

**Week 6 (Oct 23-Oct 29)**

Mo: Hamilton's estimate, type II blowups

We: Huisken's convergence theorem

**Week 7 (Oct 30-Nov 5)**

Mo: Local regularity theorem

We: Andrews' noncollapsing estimate

Sa+Su: *Minischool on Mean Curvature Flow and Ricci flow*

**Week 8 (Nov 6-Nov 12)**

Mo-Fr: *Workshop on Mean Curvature Flow and Ricci flow*

We 4pm: *Colloquium by Prof. Huisken*

**Week 9 (Nov 13-Nov 19)**

Mo: Set theoretic solutions, elliptic regularization

We: Level set solutions

We-Fr: *Coxeter Lecture series by Prof. Colding*

**Week 10 (Nov 20-Nov 26)**

Mo: geometric measure theory, Brakke solutions

We: Compactness theorem for integral Brakke flows, tangent flows

**Week 11 (Nov 27-Dec 3)**

Mo: Existence of Brakke flows via elliptic regularization

We: Ilmanen's enhanced motion

**Week 12 (Dec 4-Dec 10)**

Mo: local curvature estimate, convexity estimate, global convergence theorem

We: Regularity and structure theory for mean convex MCF