Tuesday

We will discuss our recent classification of noncollapsed singularities of the mean curvature flow in $\mathbb{R}^4$. This is joint work with Kyeongsu Choi.

Stationary integral varifolds, introduced by Almgren in the sixties, are a very useful generalization of minimal surfaces, which play an important role in a variety of geometric problems. While all known examples of nonsmooth stationary integral varifolds consist of pieces of classical minimal surfaces coming together at a codimension set of singularities, the only general regularity result available is the 1972 celebrated regularity theorem of Allard, which shows that the regular part of the varifold is dense in its support. Even proving that the singular part of 2-dimensional ones in $\mathbb R^3$ has zero $2$-dimensional measure is surprisingly challenging. In this talk I will explain what the difficulties are, propose some conjectures which we hope might simplify the problems, and present some partial results towards their solution, which anyway deliver some interesting structural consequences. The talk is based on two joint works with Camillo Brena, Stefano Decio, and Federico Franceschini.

In 1989, B. White conjectured that every Riemannian 3-sphere has at least 5 embedded minimal tori. We confirm this conjecture for 3-spheres of positive Ricci curvature. Our proof is based on a strategy of using min-max theory to imitate mean curvature flow.

On a complete non-compact Riemannian manifold, the existence of certain nontrivial harmonic functions provides a useful tool for obtaining key geometric properties through monotonic quantities. One approach to establishing monotonicity is by integrating the Bochner formula, which involves Ricci curvature, over the sub-level sets of a harmonic function. This method has been employed to derive various geometric inequalities, such as Minkowskitype inequalities for hypersurfaces, as well as volume estimates of Bishop- Gromov type. In this talk, we will survey this technique and highlight some recent applications.

The length spectrum of a Riemannian surface is a sequence of geometric invariants called p-widths, which are analogous to the eigenvalues of the Laplacian. Recent work of Chodosh and Mantoulidis guarantees that each p-width equals the length of a closed immersed geodesic. I will discuss joint work with Jared Marx-Kuo and Lorenzo Sarnataro investigating the geometric properties of these associated geodesics, including upper bounds on their Morse index and number of self-intersections.