Friday

Let X be a compact Kähler manifold. Calabi asked whether a given Kähler class on X contains a "canonical" Kähler metric, such as an extremal metric. Roughly speaking, the Yau-Tian-Donaldson conjecture states that if the Kähler class is the first Chern class of an ample line bundle, then the existence of an extremal metric should be governed by an algebro-geometric stability condition. I will present joint work with S. Boucksom, where we prove a version of this conjecture.

In this talk, we discuss recent developments in ancient solutions to the mean curvature flow in higher dimensions. Consider an ancient flow asymptotic to a cylinder with the number of R factors equal to k, we show that the asymptotic behavior of the flow is characterized by a k x k matrix Q whose eigenvalues can only be 0 and 1. We further discuss the cases where Q is fully degenerate or fully nondegenerate under the noncollapsing assumption. In the fully degenerate case, we obtain a complete classification. In the fully nondegenerate case, we establish a rigidity result showing that the solutions are determined by only k-1 parameters. This is based on joint work with Beomjun Choi and Wenkui Du.

Let $M$ be a closed $3$-dimensional Riemannian manifold with positive scalar curvature, $R_g \ge 6$. We show that $M$ contains a non-trivial closed geodesic of length less than $22,500$.

We report on joint work with Tamás Darvas, in which we characterize geodesic lines in the (completed) space of Kähler metrics in terms of their Legendre transform. Using this, we construct numerous geodesic lines on any projective manifold, which are not generated by holomorphic vector fields in general; this disproves a folklore conjecture popularized by Berndtsson. In the case of Riemann surfaces, our results can be significantly sharpened. Finally, we investigate the validity of Euclid's fifth postulate for the space of Kähler metrics.