UC San Diego Geometry Seminar 2020-2021

Unless otherwise noted, all seminars are on Wed 11:00 - 12:00 in Zoom setting. Zoom ID: 960 7952 5041

Fall 2020 Schedule

October 7, 2020, Zoom ID: 960 7952 5041

Salvatore Stuvard, U. Texas (Joint with Stanford)
Title: On the Brakke flow of surfaces with fixed boundary conditions
Abstract: Brakke flow is a measure-theoretic generalization of the mean curvature flow which exploits the flexibility of geometric measure theory in order to describe the evolution by (generalized) mean curvature of surfaces exhibiting singularities, such as, for instance, a planar network with multiple junctions. In the first part of this talk, I will discuss the proof of the following result: given any $n$-dimensional rectifiable subset $\Gamma_0$ of a strictly convex bounded domain $U \subset \mathbb{R}^{n+1}$ such that $U \setminus \Gamma_0$ is not connected, there exists a Brakke flow of surfaces (possibly weighted with integer multiplicities) starting from $\Gamma_0$ and with the additional property that their boundary coincides with $\partial \Gamma_0$ at all times. Furthermore, the flow subconverges, as $t \to \infty$ and in the sense of varifolds, to a (generalized) minimal surface in $U$ with the prescribed boundary $\partial \Gamma_0$, thus providing a dynamical solution to Plateau's problem. In the second part, I will discuss recent developments concerning the relationship between the singularities of a stationary initial surface $\Gamma_0$ and the (non) uniqueness of the flow. This investigation leads to the definition of a class of \emph{dynamically stable} stationary varifolds, which seems worthy of further study. Based on joint works with Yoshihiro Tonegawa (Tokyo Institute of Technology).

October 14 , 2020

Elia Brue
Title: Boundary regularity and stability for spaces with Ricci curvature bounded below
Abstract: The theory of RCD spaces has seen a huge development in the last teen years. They are metric measure structures satisfying a synthetic notion of Ricci bounded below. This class includes several spaces with boundary, such as Gromov-Hausdorff limits of manifolds with convex boundary and Ricci bounded below in the interior. In this talk we will present new stability and regularity results for boundaries of RCD spaces. We will focus mostly on a new epsilon-regularity theorem which is new even in the setting of smooth Riemannian manifolds. It is based on a work in progress joint with Aaron Naber and Daniele Semola.

October 21, 2020

Pak-Yeung, Chan
Title: Steady Kaehler Ricci soliton with nonnegative Ricci curvature and integrable scalar curvature
Abstract: Ricci soliton is a self-similar solution to the Ricci flow and arises naturally in the singularity analysis of the flow. Steady Ricci soliton is a kind of soliton whose associated Ricci flow evolves by reparametrizing a fixed metric. It is closely related to the Type II limit solution to the Ricci flow. Steady Ricci soliton with integrable scalar curvature was studied by Deruelle in 2012, later by Catino-Mastrolia-Monticelli in 2016, Munteanu-Sung-Wang in 2019, Deng-Zhu in 2020. In this talk, we shall discuss a classification result on steady Kaehler Ricci soliton with nonnegative Ricci curvature and integrable scalar curvature. We then apply the result to study the steady Kaehler Ricci soliton with subquadratic volume growth or fast curvature decay.

October 28, 2020

Xavier Fernandez-Real Girona
Title: The non-regular part of the free boundary for the fractional obstacle problem
Abstract: The fractional obstacle problem in $\mathbb R^n$ with obstacle $\varphi\in C^\infty(\mathbb{R}^n)$ can be written as \[ \min\{(-\Delta)^s u , u-\varphi\} = 0,\quad\textrm{in }\quad\mathbb{R}^n. \] The set $\{u = \varphi\} \subset \mathbb{R}^n$ is called the contact set, and its boundary is the free boundary, an unknown of the problem. The free boundary for the fractional obstacle problem can be divided between two subsets: regular points (around which the free boundary is smooth, and is $n-1$ dimensional) and degenerate points. The set of degenerate points, even for smooth obstacles, can be very large (for example, with infinite $\mathcal{H}^{n-1}$ measure). In a joint work with X. Ros-Oton we show, however, that generically solutions to the fractional obstacle problem have a lower dimensional degenerate set. That is, for almost every solution (in an appropriate sense), the set of degenerate points is lower dimensional.

November 4, 2020

Man-Chun Lee, Northwestern
Title:Kahler-Ricci flow with unbounded curvature and application
Abstract: In this talk, we will discuss the existence theory of Kahler-Ricci flow when the Kahler metric has unbounded curvature. We will discuss some application of the Kahler-Ricci flow in the study of uniformization and the regularity of Gromov-Hausdorff limit. This is joint work with L.-F. Tam.

November 11, 2020

Damin Wu, UConn
Title: Bergman metric on complete Kahler manifold.
Abstract:It has been proved by R. E. Greene and H. Wu that a simply-connected complete Kahler manifold negatively pinched sectional curvature possesses a complete Bergman metric. I will briefly review the history and present the estimates of the Bergman metric using the bounded geometry. This talk is based on the joint work with Yau.

November 18, 2020

Natasa Sesum, Rutgers
Title: Ancient solutions in geometric flows
Abstract: We will discuss classification of ancient solutions to geometric flows. We will focus especially on the Ricci flow.

December 2, 2020

Gabor Szekelyhidi, Notre Dame
Title: Uniqueness of certain cylindrical tangent cones
Abstract: Leon Simon showed that if an area minimizing hypersurface admits a cylindrical tangent cone of the form C x R, then this tangent cone is unique for a large class of minimal cones C. One of the hypotheses in this result is that C x R is integrable and this excludes the case when C is the Simons cone over S^3 x S^3. The main result in this talk is that the uniqueness of the tangent cone holds in this case too. The new difficulty in this non-integrable situation is to develop a version of the Lojasiewicz-Simon inequality that can be used in the setting of tangent cones with non-isolated singularities.

Additional talk at 3:00pm

Quoc-Hung Nguyen

December 9, 2020

Albert Chau, UBC

Spring 2020 Schedule

Questions: lni@math.ucsd.edu