Math 646, Spring 2009

Professor Minicozzi

Lectures:        1:30 to 2:50 Mondays and Wednesdays in Maryland 309. 

Overview:    This is a second semester course in Riemannian geometry.  The basic reference will be Do Carmo's Riemannian Geometry.  The first topic of the course will be the paper "Width and mean curvature flow" by Colding and Minicozzi in volume 12, number 5, of Geometry and Topology.

 Overview of the lectures:

Lecture 1:  Existence of minimizing segments; classification of surfaces; minimizing in a free homotopy class; geodesics as critical points of the length functional; second variation and index; great circles have index 1.

Lecture 2: Degree of a map and Pi_2 of S^2; Sweepouts of S^2 and the width; the W^{1,2} topology on piecewise linear curves.

Lecture 3: Birkhoff's curve shortening process (BCSP); Existence of good sweepouts where almost maximal curves are almost geodesics; existence of unstable geodesics on S^2 using the key properties of BCSP.

Lecture 4: Mean curvature flow; Huisken's theorem for convex hypersurfaces; the width and mean curvature flow.

Lecture 5: Completion of the proof of the estimate for the width under mean curvature flow.

Lecture 6: Proof of the properties of BCSP: properties 1, 3, and 4 on page 2520 of "Width and mean curvature flow".

Lecture 7: Completion of 1, 3, and 4.

Lecture 8: Property 2 of BCSP (continuity with respect to the curve).

Lecture 9: Harmonic maps from S^2: the Hops differential and conformality; harmonic map heat flow.