Gluing of Geometric PDEs - Obstructions vs. Constructions for Minimal Surfaces & Mean Curvature Flow Solitons

Lecture by Niels Martin Møller, Princeton

Abstract:

For geometric nonlinear PDEs, where no easy superposition principle
holds, examples of (global, geometrically/topologically interesting)
solutions can be hard to come about. In certain situations, for
example for 2-surfaces satisfying an equation of mean curvature type,
one can generally "fuse" two or more such surfaces satisfying the PDE,
as long as certain global obstructions are respected - at the cost (or
benefit) of increasing the genus significantly. The key to success in
such a gluing procedure is to understand the obstructions from a more
local perspective, and to allow sufficiently large geometric
deformations to take place.
  In the talk I will introduce some of the basic ideas and techniques
(and pictures) in the gluing of minimal 2-surfaces in a 3-manifold.
Then I will explain two recent applications, one to the study of
solitons with genus in the singularity theory for mean curvature flow
(Ilmanen's conjectured "planosphere" self-shrinkers), and another to
the non-compactness of moduli spaces of finite total curvature minimal
surfaces (a problem posed by Ros & Hoffman-Meeks). Some of the work is
joint w/ Steve Kleene and/or Nicos Kapouleas.