Joint Geometric Analysis Seminar 
 (Part of MIST program)

  • Organizers: Man Chun Lee, Conan Leung and Martin Li
  • Hosts: Department of Mathematics & Institute of Mathematical Sciences, CUHK
  • Contact: Please email me at martinli@math.cuhk.edu.hk if you would like to contribute a talk.


  • Remark: Due to current COVID-19 situation, we will be hosting the seminar talks online via ZOOM until safe travels are permitted. You can access the talk via the ZOOM link below.


Upcoming talk


Date/Time/Venue Talks
Nov 25, 2022 (Fri)
11:00AM
Zoom link		 Speaker: Ngoc Cuong Nguyen (Korea Advanced Institute of Science & Technology)
Title: The Dirichlet problem for the Monge-Ampere equation on Hermitian manifold with boundary

Abstract:
This is joint work with S. Kolodziej. We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Ampère equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and Hölder continuous quasi-plurisubharmonic functions. The continuity of the solution is proved for measures that well dominated by capacity, for example measures with $L^p$, p>1, densities, or moderate measures in the sense of Dinh-Nguyen-Sibony.


2022-23 Fall schedule

Date/Time/Venue Talks
Sep 30, 2022 (Fri)
11:00AM
Zoom link		 Speaker: Shouhei Honda (Tohoku University)
Title: Singular Weyl’s law with Ricci curvature bounded below

Abstract:
In this talk we discuss unexpected singular asymptotic behaviors of eigenvalues on some compact Ricci limit, in particular RCD, spaces. This is a joint work with Xianzhe Dai (UC Santa Barbara), Jiayin Pan (UC Santa Cruz) and Guofang Wei (UC Santa Barbara).
Oct 7, 2022 (Fri)
11:00AM
Zoom link		 Speaker: Adrian Chun-Pong Chu (University of Chicago)
Title: A free boundary minimal surface via a 6-sweepout

Abstract:
We will establish the following Morse theoretic result. We study the topology of the space of surfaces with genus 0 or 1 in the Euclidean unit 3-ball, and then find a free boundary minimal surface with genus, Morse index, and area control via min-max theory.
Oct 14, 2022 (Fri)
11:00AM
Zoom link		 Speaker: Douglas Stryker (Princeton University)
Title: Volume growth and first eigenvalue of noncompact 3-manifolds with positive scalar curvature

Abstract:
Gromov's μ-bubble technique is a powerful tool in 3-manifold geometry. I will present a general geometric argument using μ-bubbles that applies to 3-manifolds with positive scalar curvature and stable minimal hypersurfaces in certain 4-manifolds, based on joint work with Otis Chodosh and Chao Li.
Oct 28, 2022 (Fri)
3:00PM
Zoom link		 Speaker: Shih-Kai Chiu (University of Oxford)
Title: Higher regularity for singular Kähler-Einstein metrics

Abstract:
In this talk we consider singular Kähler-Einstein metrics that are obtained as Gromov-Hausdorff limits of polarized Kähler-Einstein manifolds. We first show that when the metric tangent cone at a point is isomorphic to the germ of the singularity, then the singular metric converges to its tangent cone at a polynomial rate on the level of Kähler potentials. When the tangent cone has a smooth cross section, this implies polynomial convergence in the usual sense, generalizing a result of Hein-Sun. We also obtain a similar result for a class of examples when the tangent cone is not isomorphic to the germ of the singularity. Finally, similar techniques allow us to prove a rigidity result for complete ddbar-exact Calabi-Yau metrics with maximal volume growth. This talk is based on joint work with Gábor Székelyhidi.
Nov 4, 2022 (Fri)
11:00AM
Zoom link		 Speaker: Eric Chen (University of California, Berkeley)
Title: The Yamabe flow on asymptotically Euclidean manifolds

Abstract:
While on compact manifolds, the Yamabe flow generally converges to a metric of constant scalar curvature, long-time existence or convergence of the flow does not always hold on noncompact manifolds. I will discuss the behavior of the Yamabe flow on asymptotically Euclidean manifolds. In this case, long-time existence of the Yamabe flow always holds, and the flow converges if and only if the Yamabe constant of the initial metric's conformal class is positive. When convergence fails in the case of a nonpositive Yamabe constant, the blowup profile at time infinity can be described using the solution of the Yamabe problem on a singular compactification of the original manifold. This is joint work with Gilles Carron and Yi Wang.
Nov 11, 2022 (Fri)
9:00AM
Zoom link		 Speaker: Paula Burkhardt-Guim (Courant Institute, New York University)
Title: ADM mass for $C^0$ metrics and distortion under Ricci-DeTurck flow

Abstract:
We show that there exists a quantity, depending only on $C^0$ data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the $C^0$ sense and has nonnegative scalar curvature in the sense of Ricci flow. Moreover, the $C^0$ mass at infinity is independent of choice of $C^0$-asymptotically flat coordinate chart, and the $C^0$ local mass has controlled distortion under Ricci-DeTurck flow when coupled with a suitably evolving test function.
Nov 18, 2022 (Fri)
3:00PM
Zoom link		 Speaker: John Man-shun Ma (University of Copenhagen)
Title: Entropy bounds, Compactness and Finiteness Theorems for Embedded self-shrinkers with rotational symmetry

Abstract:
In this talk, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in $R^{n+1}$. First, using comparison geometry in the context of metric geometry, we derive explicit upper bounds for the entropy of all such self-shrinkers. Second, as an application we prove a smooth compactness theorem on the space of all such shrinkers. We also prove that there are only finitely many such self-shrinkers with an extra reflection symmetry. This is a joint work with A. Muhammad and N. M. Moeller.
Nov 25, 2022 (Fri)
11:00AM
Zoom link		 Speaker: Ngoc Cuong Nguyen (Korea Advanced Institute of Science & Technology)
Title: The Dirichlet problem for the Monge-Ampere equation on Hermitian manifold with boundary

Abstract:
This is joint work with S. Kolodziej. We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Ampère equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and Hölder continuous quasi-plurisubharmonic functions. The continuity of the solution is proved for measures that well dominated by capacity, for example measures with $L^p$, p>1, densities, or moderate measures in the sense of Dinh-Nguyen-Sibony.





Past Seminars



© Martin Li, Department of Mathematics, The Chinese University of Hong Kong