MATHEMATICS 667- MODERN GEOMETRY- SPRING 2008

INTRODUCTION TO GEOMETRIC EVOLUTION EQUATIONS

Math 667 announcement

Course outline and references

course handouts

1/10 Mean curvature flow: geometry of hypersurfaces

(see handout 1)

Huisken 's 1984 paper on mean curvature flow for convex hypersurfaces: see handout 2

INTRODUCTION TO MEAN CURVATURE FLOW

(first handout) 1/15, 17 (first variation formula)

EVOLUTION OF THE GEOMETRY UNDER MEAN-CURVATURE FLOW

(second handout) 1/22, outline of Huisken's 1984 JDG paper

1/24: Maximum principles and applications to MCF (source: Chow/Knopf, ch.4)

1/29: Proof of Hamilton's maximum principle; discussion of strong maximum principles

1/31: Local existence for MCF in Hoelder spaces (statement). Global existence (|A|^2 blows up at the

critical time), beginning of proof: evolution of covariant derivatives of A and their norms, a maximum principle.

2/12: The mean curvature ratio tends to 1; the flow normalized to constant area.

2/14: class cancelled

2/19: area-normalized MCF: upper/lower mean curvature bounds, infinite existence time, pointwise curvature pinching

Type I singularities, rescaling to bounded A/ Huisken's monotonicity formula (start)

2/29, 3/4, 3/6: short-time existence for Ricci flow

NON-ELLIPTIC STRUCTURE OF THE RICCI OPERATOR

(third handout)