UCSD Differential Geometry Seminar 2023

1:00 pm - 2:00pm at APM 5829


Spring 2023 Schedule

April 6

Daniel Stern, Chicago
Title: Existence theory for harmonic maps and connections to spectral geometry
Abstract: I’ll discuss recent progress on the existence theory for harmonic maps, in particular the existence of harmonic maps of optimal regularity from manifolds of dimension n>2 to every non- aspherical closed manifold containing no stable minimal two-spheres. As an application, we’ll see that every manifold carries a canonical family of sphere-valued harmonic maps, which (in dimension<6) stabilize at a solution of a spectral isoperimetric problem generalizing the conformal maximization of Laplace eigenvalues on surfaces. Based on joint work with Mikhail Karpukhin.

April 13,

Jiangtao Li, UCSD
Title: Yau's conjecture on the first eigenvalue
Abstract: This talk is an exposition on the partial progress of Yau's conjecture on the lower estimate of first eigenvalue of embedded minimal hypersurfaces in a unit sphere.

April 20


Pak Yeung Chan, UCSD
Title: Curvature and gap theorems of gradient Ricci solitons
Abstract:Ricci flow deforms the Riemannian structure of a manifold in the direction of its Ricci curvature and tends to regularize the metric. This provides useful information about the underlying space. Ricci solitons are special solutions to the Ricci flow and arise naturally in the singularity analysis of the flow. We shall discuss some curvature and entropy gap theorems of gradient Ricci solitons. This talk is based on joint works with Yongjia Zhang and Zilu Ma, Eric Chen and Man-Chun Lee.

April 27



Title:
Abstract:

May 4


Jonathan Zhu, Univ. Washington
Title: Distance comparison principles for curve shortening flows
Abstract: For closed curves evolving by their curvature, the theorem of Gage-Hamilton and Grayson establishes that an embedded curve contracts to a round point. An efficient proof was later found by Huisken, with improvements by Andrews-Bryan, which uses multi-point maximum principle techniques. We’ll discuss the use of these techniques in other settings, particularly for the long-time behaviour of curve shortening flow with free boundary.

May 11


Songying Li, UCI
Title: Bergman metric with constant curvature and uniformization theorems
Abstract: This talk is based on a joint work with Xiaojun Huang entitled “Bergman metrics as pull-backs of the Fubini-Study metric”. We study domains in $C^n$ or Stein manifolds M such that their Bergman metrics have constant holomorphic sectional curvature κ. We prove a uniformization theorem when κ < 0 through the Calabi rigidity theorem and holomorphic extension theorems. We also discuss the case when κ ≥ 0. We provide several interesting examples of existence of such M. Under certain conditions on M, we prove that the Bergman metric of M can not have non-negative constant holomorphic sectional curvatures.

May 18


Ronan Conlon, UT Dallas
Title: Shrinking Kahler-Ricci solitons
Abstract: Shrinking Kahler-Ricci solitons model finite-time singularities of the Kahler-Ricci flow, hence the need for their classification. I will talk about the classification of such solitons in 4 real dimensions. This is joint work with Deruelle-Sun, Cifarelli-Deruelle, and Bamler-Cifarelli-Deruelle.

May 25,


Hongyi Shen, UCSD
Title:Deformations of the Scalar Curvature and the Mean Curvature
Abstract: On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. We prove that this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary, as motivated by an attempt to generalize the Riemannian Penrose inequality in dimension 8. This result is a generalization of Corvino's result about localized scalar curvature deformations; however, the existence part needs to be handled delicately since the problem is non-variational. For non-generic cases, we give a classification theorem for domains in space forms and Schwarzschild manifolds, and show the connection with positive mass theorems.

June 1


Liao Yuan, UCSD
Title: Homogenous structures and the Ricci flow
Abstract: This talk will focus on Böhm and Lafuente 2017's work on immortal homogeneous Ricci flow, where they prove that any sequence of blow-downs of such flow will subconverges to an expanding homogeneous Ricci soliton. For a \mathfrac{g}-homogeneous space M, Ricci flow of G-invariant metrics can be shown to be equivalent to a flow on Ad(H) invariant "bracket." We will show the existence of stratification of the space of brackets that induces curvature estimate on each strata, motivated by geometric invariant theory. The sharp case of the inequality corresponds to the limit case of an expanding G-invariant Ricci soliton, and we will show that as immortal homogeneous Ricci flow is of Type III, the blowdown will subconverges to such limit case. Finally if time allows, we will discuss Böhm and Lafuente recent proof of Alekseevskii's conjecture and its implication to homogeneous Ricci soliton.

June 8


Davide Parise, UCSD
Title: A gauge-theoretic construction of codimension-two mean curvature flows.
Abstract: Mean curvature flow is the negative gradient flow of the area functional, and it has attracted a lot of interest in the past few years. In this talk, we will discuss a PDE-based, gauge theoretic, construction of codimension-two mean curvature flows based on the Yang-Mills-Higgs functionals, a natural family of energies associated to sections and metric connections of Hermitian line bundles. The underlying idea is to approximate the flow by the solution of a parabolic system of equations and study the corresponding singular limit of these solutions as the scaling parameter goes to zero. This is based on joint work with A. Pigati and D. Stern.

Winter 2021 Schedule

Autumn 2020 Schedule

Questions: leni@ucsd.edu