《MSC_Colloq_020811》.pdfVIP

Geometric heat equations and the sphere theorem Ben Andrews Mathematical Sciences Center, Tsinghua University Mathematical Sciences Institute, Australian National University Morningside Center of Mathematics, Chinese Academy of Sciences Tsinghua University, August 2, 2011 2 Outline of the talk: A quick introduction to the heat equation A quick introduction to manifolds and curvature Global differential geometry and the sphere theorem A quick introduction to heat flows of hypersurfaces Proving the sphere theorem with hypersurface flows Exotic spheres A quick introduction to Ricci flow The differentiable sphere theorem A quick primer on heat equations 3 My first task is to explain why it is a good idea to use heat equations. Consider the classical heat equation, in which a heat distribution ux 1xn t evolves in time according to the equation u n 2 u t u x x i1 i i % To make sense of this (so a unique solution exists for any initial heat distribu- % tion) we need boundary conditions. For simplicity consider spatially periodic conditions: ux z t ux t for any x and t and any z in some lattice (for n % example, the integer lattice in ). In effect we are looking at the heat equa- tion defined on a torus, so there are no boundary issues to worry about. % I want to point out some features of this simple equation which carry over to much more general ‘heat-type’ or ‘parabolic’ equations: (1). Maximum principle: At a point where u attains a maximum in x , the second derivatives in each direction are