Workshop

Zoom Link

In this talk, I will discuss the problem of when an Alexandrov space is smoothable. I will review the history of this question and discuss a new result that partially answers it. This is joint work with Pedro Solórzano and Fred Wilhelm.

Zoom Link

Steady Ricci solitons are fundamental objects in the study of Ricci flow, as they are self-similar solutions and often arise as singularity models. Classical examples of steady solitons are the most symmetric ones, such as the 2D cigar soliton, the O(n)-invariant Bryantsolitons, and Cao’s U(n)-invariant Kahler steady solitons. Recently we constructed a family of flying wing steady solitons in any real dimension n≥3, which confirmed a conjecture by Hamilton in n=3. In dimension 3, we showed all steady gradient solitons are O(2)-symmetric. In the Kahler case, we also construct a family of Kahler flying wing steady gradient solitons with positive curvature for any complex dimension n≥2. This answers a conjecture by H.-D.Cao in the negative. This is partly collaborated with Pak-Yeung Chan and Ronan Conlon.

Zoom Link

In a joint work with Vanderson Lima (UFRGS, Brazil), we introduced a family of functionals on the space of Riemannian metrics of a compact surface with boundary, defined via eigenvalues of a Steklov-type problem. In this talk we will prove that each such functional is uniformly bounded from above, and we will characterize maximizing metrics as induced by free boundary minimal immersions in some geodesic ball of a round sphere. Also, we will determine that the maximizer in the case of a disk is a spherical cap of dimension two, and we will prove rotational symmetry of free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions.

Zoom Link

Zoom Link

We show that for every $n\geq 4$ and every $\epsilon >0$ there exists a closed manifold $M$ which admits a metric of curvature contained in the interval $[-1-\epsilon,-1+\epsilon]$, which admits a negatively curved Einstein metric but no metric of constant curvature.

Zoom Link

Poincare-Einstein (PE) manifolds such as Poincare ball model are complete Einstein manifolds with a well-defined conformal boundary. There is a rich interplay between the conformal geometry of the boundary of a PE manifold and the Riemannian geometry of its interior. A first step in studying the moduli space of PE manifolds is to develop a good understanding of its global invariants. In even dimensions, renormalized curvature integrals give many such invariants. In this talk, I will discuss a general procedure for computing renormalized curvature integrals on PE manifolds. In particular, this explains the connection between the Gauss-Bonnet-type formulas of Albin and Chang-Qing-Yang for the renormalized volume, and explicitly identify a scalar conformal invariant in the latter formula. Our procedure also produces similar formulas for compact Einstein manifolds. This talk is based on joint works with Jeffrey Case, Ayush Khaitan, Aaron Tyrrell, and Wei Yuan.

Zoom Link

In this talk we construct explicit examples that show the sublevel sets of the solution of a k-Hessian equation defined on a convex ring do not have to be convex.

Zoom Link

In this talk, I will discuss existing results and open problems about estimating the Morse index of a (free boundary) minimal surface from below by a function of its topology.
I will mostly focus on results proving that the Morse index of a free boundary minimal surface in a three-dimensional Riemannian manifold grows linearly with the product of its area and its topology. This is joint work with Santiago Cordero-Misteli, and it is inspired by Antoine Song's result in the closed case.