The moduli spaces of two-convex embedded spheres and tori
It is interesting to study the topology of the space of smoothly embedded n-spheres in Rn+1. By Smale’s theorem, this space is contractible for n=1 and by Hatcher’s proof of the Smale conjecture, it is also contractible for n=2. These results are of great importance, generalising in particular the Schoenflies theorem and Cerf’s theorem. In this talk, I will explain how mean curvature flow with surgery can be used to study a higher-dimensional variant of these results, proving in particular that the space of two-convex embedded spheres is path-connected in every dimension n. We then also look at the space of two-convex embedded tori where the result in particular depends on the dimension n. This is all joint work with Robert Haslhofer and Or Hershkovits.