We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal surfaces with bounded index on a given three-manifold might degenerate. We then discuss several applications, including some compactness results. Time permitting, we discuss how our strategy can be extended to ambient dimensions 4,5,6 and 7. (This is joint work with O. Chodosh and D. Ketover) 

Talk Schedule

2:00 - 2:50 PM     Karen Uhlenbeck (UT Austin)

3:10 - 4:00 PM     Gudlaugur Thorbergsson (Cologne)

4:00 - 4:30 PM     Reception

4:30 - 5:20 PM     Robert Bryant (Duke)

We prove that Alexandrov's conjecture relating the area and diameter of a convex surface holds for the surface of a general ellipsoid. This is a direct consequence of a more general result which estimates the deviation from the optimal conjectured bound in terms of the length of  the cut locus of a point on the surface. We also prove that the natural extension of the conjecture to general dimension holds among closed convex spherically symmetric Riemannian manifolds. Our results are based on a new symmetrisation procedure which we believe to be interesting in its own right.

This is joint work with Pedro Freitas accepted for publication in the Tohoku Mathematical Journal, preprint on http://arxiv.org/abs/1406.0811.

We will explain a systematic way to study a type of curve flows in R^{n+1, n}, whose geometric invariants are solutions of some integrable systems. In detail, we will construct a hierarchy of isotropic curve flows in R^{2, 1}, construct explicit solutions from Backlund transformation, and study its Hamiltonian formulation.