Kodaira Dimension and the Yamabe Problem, Revisited

  • Speaker: Claude LeBrun
  • Time: 9:00-10:00
  • Abstract: Dimension four provides a surprisingly idiosyncratic setting for the interplay between scalar curvature and differential topology. This peculiarity becomes especially pronounced when discussing the Yamabe invariant or “sigma constant” of a smooth compact manifold; and Seiberg-Witten theory makes this especially apparent for those 4-manifolds that arise as compact complex surfaces. For compact complex surfaces of Kaehler type, I showed in the late 1990s that the sign of the Yamabe invariant is always determined by the Kodaira dimension, and calculated the Yamabe invariant exactly in all cases where it is non-positive. In this talk, I will describe recent joint work with Michael Albanese that generalizes these results to all complex surfaces of non-Kaehler type, except those of class VII. However, I will also explain why excluding this exceptional class is absolutely essential for these purposes.

Ricci flows with nonstandard initial data

Most Ricci flow theory takes the short-time existence of solutions as a starting point and ends up concerned with understanding the long-time limiting behaviour and the structure of any finite-time singularities that may develop along the way. In this talk I will look at what you can think of as singularities at time zero. I will describe some of the situations in which one would like to start a Ricci flow with a space that is something other than a smooth bounded curvature Riemannian manifold, and some of the situations in which one considers smooth initial data that is only achieved in a non-smooth way. A particularly interesting and useful case is the problem of starting a Ricci flow on a Riemann surface equipped with a measure, as I will explain.
Parts of the talk are joint with either Hao Yin (USTC) or Man Chun Lee (CUHK).

Symmetry of Plateau-minimal surfaces

  • Speaker: Jacob Bernstein
  • Time: 11:20-12:20
  • Abstract: Using the method of moving planes Schoen showed that any smooth minimal immersion spanning a pair of coaxial circles must be a piece of a catenoid or a pair of disks. Using a variation on this technique we extend this result to a larger class of stationary surfaces -- at least when the circles have the same radius. This class is large enough to include the essentially singular surfaces that are observed in physical soap-films. This is joint work with F. Maggi.

ALG spaces and the Hitchin equations (via Zoom)

  • Speaker: Rafe Mazzeo
  • Time: 2:00-3:00
  • Abstract: The natural metric on the moduli space of solutions to the Hitchin equations on a Riemann surface is hyperkaehler. On the four-punctured sphere, this moduli space is in fact a 4-dimensional gravitational instanton of ALG type. A long-standing question asks whether all gravitational instantons arise as gauge-theoretic moduli spaces (or similar finite dimensional analogues). I will explain how and why this is true in this particular setting, namely that the available parameters in the Hitchin construction exhaust the full moduli of ALG metrics of D4 type. Joint with Fredrickson, Swoboda, Weiss.

O(2)-symmetry of 3D steady gradient Ricci solitons (via Zoom)

  • Speaker: Yi Lai
  • Time: 3:10- 4:10
  • Abstract: For any 3D steady gradient Ricci soliton, if it is asymptotic to a ray we prove that it must be isometric to the Bryant soliton. Otherwise, it is asymptotic to a sector and called a flying wing. We show that all flying wings are O(2)-symmetric. Hence, all 3D steady gradient Ricci solitons are O(2)-symmetric.

Quantitative Faber-Krahn Inequalities and Applications

Among all drum heads of a fixed area, a circular drum head produces the vibration of lowest frequency. The general dimensional analogue of this fact is the Faber-Krahn inequality, which states that balls have the smallest principal Dirichlet eigenvalue among subsets of Euclidean space with a fixed volume. I will discuss new quantitative stability results for the Faber-Krahn inequality on Euclidean space, the round sphere, and hyperbolic space, as well as an application to the Alt-Caffarelli-Friedman monotonicity formula used in free boundary problems. This is based on joint work with Mark Allen and Dennis Kriventsov.

Immersed mean curvature flows with non-collapsed singularities

  • Speaker: Keaton Naff
  • Time: 5:30-6:30
  • Abstract: In the mean curvature flow of hypersurfaces, noncollapsing has proven to be a powerful and useful assumption when studying singularities and high curvature regions. In particular, the assumption of noncollapsing has been used to prove a wide range of local a priori estimates and has led to classification results for certain classes of singularity models. Less is known for immersed mean-convex flows. In this talk, I would like to survey recent results and discuss outstanding conjectures for immersed mean-convex flows that begin to bridge the gap between the embedded and immersed mean-convex settings. The talk is based on joint work with S. Brendle and ongoing work with S. Lynch.