Reto Buzano: Mean curvature flow and mean-convex embeddings in three-manifolds

In this talk we will study the space of mean-convex embeddings of spheres and tori into three dimensional manifolds. After motivating the problem, we first start with a brief and intuitive overview of mean curvature flow with surgery and explain a gluing construction to topologically undo the surgeries again. We then use this surgery and gluing approach to prove that the moduli space of mean convex two-spheres embedded in complete, orientable 3-dimensional manifolds with nonnegative Ricci curvature is path-connected. This result is sharp in the sense that neither of the conditions of (strict) mean convexity, completeness, and nonnegativity of the Ricci curvature can be dropped or weakened. We next study the number of path components of mean convex Heegaard tori, again in ambient manifolds with nonnegative Ricci curvature. We prove that there are always either one or two path components and this number does not only depend on the homotopy type of the ambient manifold. We give a precise characterisation of the two cases and also discuss what happens if the mean convexity condition is weakened to nonnegative mean curvature. This is joint work with Sylvain Maillot, building on earlier joint work with Robert Haslhofer and Or Hershkovits.