The moduli space of two-convex embedded spheres
  We prove that the moduli space of 2-convex embedded n-spheres in R^{n+1} is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques' influential proof of the path-connectedness of the moduli space of positive scalar curvature metics on three-manifolds. 
Reto Buzano, Robert Haslhofer, Or Hershkovits
math.DG math.GT