The moduli space of two-convex embedded spheres and tori

Reto Buzano, Queen Mary University of London. Part of the geometry seminar series.

It is interesting to study the topology of the space of smoothly embedded n-spheres in R^{n+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 2-convex embedded spheres is path-connected in every dimension n. We then also look at the space of 2-convex embedded tori where the question is more intriguing and the result in particular depends on the dimension n. This is all joint work with Robert Haslhofer and Or Hershkovits.

Reto Buzano, Queen Mary University of London