Titel: Multi-point maximum principles and variational methods in
geometric heat equations
Lecture 1: Moduli of continuity and applications to eigenvalue inequalities
In this lecture I will use similar arguments (in particular, developing maximum principles for functions involving several points) to control the modulus of continuity for solutions of the heat equation in various contexts, deriving sharp inequalities on eigenvalues as a consequence.
In particular I will describe my proof with Julie Clutterbuck of the fundamental gap conjecture for convex Euclidean domains.
Lecture I Notes
Lecture 2: Isoperimetric methods for the curve-shortening flow and for
the Ricci flow on surfaces
I will show how ideas of Hamilton can be strengthened to give very strong control on the isoperimetric profile of a solution of Ricci flow on the two-dimensional sphere, directly implying exponential convergence to a constant curvature metric. Similarly, previous estimates of Hamilton and Huisken can be improved to give a remarkably direct proof of Grayson’s theorem, that embedded convex curves in the plane become circular in shape as they contract to points under the curve shortening flow.
Lecture II Notes
Lecture 3: Non-collapsing in mean curvature flow and the Lawson and
In this lecture I will show how similar multi-point maximum principles can be applied to the mean curvature flow of hypersurfaces, and prove a `non-collapsing’ estimate for mean-convex solutions. I will briefly describe some of the applications of this estimate in the analysis of mean curvature flow, and perhaps
some extensions to a wider family of evolution equations.
Lecture III Notes
Lecture 4: Gauss curvature flows and entropy functionals
In the final lecture I will describe some recent progress in understanding the motion of hypersurfaces by their Gauss curvature. Here the methods are based around the analysis of a family of `entropies’ associated with these flows, and the techniques are quite different from previous analyses of hypersurface flows.
In particular there are some interesting connections with geometric inequalities
arising in the theory of convex bodies and affine differential geometry.
Title: "Curvature flows with surgery"
Themes: Mean curvature flow with surgery (Introduction), Estimates based on the maximum principle, integral estimates, monotonicity formula, pseudo-locality, non-collapsing, classification of singularities, long-time existence, new results on the mean curvature flow of mean convex two dimensional surfaces (Huisken and Brendle), surgery for fully nonlinear flows.
- G. Huisken and C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, Invent. Math. 175, 137–221 (2009)
- Mean curvature flow with surgery of mean convex surfaces in R^3 S. Brendle, G. Huisken http://arxiv.org/abs/1309.1461
- G. Huisken and C. Sinestrari, Convex ancient solutions of the mean curvature flow, Journal Differential Geometry, 101 (2015), no. 2, 267–287.
- G. Huisken and S. Brendle, A fully nonlinear flow for 2-convex hypersurfaces, 41pp., (2015), arXiv:1507.04651.