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
TBA
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Zoom link		 Speaker: TBA
Title: TBA

Abstract:
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2021-22 Spring schedule

Date/Time/Venue Talks
Jan 12, 2022 (Wed)
10:00AM
Zoom link		 Speaker: Jonathan Zhu (Princeton University)
Title: Min-max theory for capillary surfaces

Abstract:
Capillary surfaces model interfaces between incompressible immiscible fluids. The Euler-Lagrange equations for the capillary energy functional reveals that such surfaces are solutions of the prescribed mean curvature equation, with prescribed contact angle where the interface meets the container of the fluids. Min-max methods have been used with great success to construct unstable critical points of various energy functionals, particularly for the special case of closed minimal surfaces. We will discuss the development of min-max methods to construct general capillary surfaces.
Jan 19, 2022 (Wed)
10:00AM
Zoom link		 Speaker: Ao Sun (University of Chicago)
Title: Existence of minimal hypersurfaces with arbitrarily large area

Abstract:
I will present an approach to find minimal hypersurfaces with arbitrarily large area in a closed manifold with dimension between 3 and 7. The method is based on the novel Almgren-Pitts min-max theory, and its further development by Marques-Neves, Song and Zhou. Among the applications, we can show that there exist minimal hypersurfaces with arbitrarily large area in an analytic manifold. In the case where this approach does not work, it is surprising that the space of minimal hypersurfaces has a Cantor set fractal structure.This is joint work with James Stevens (UChicago).
Jan 26, 2022 (Wed)
10:00AM
Zoom link		 Speaker: Siyuan Lu (McMaster University)
Title: Rigidity of Riemannian Penrose inequality with corners and its implications

Abstract:
Motivated by the rigidity case in the localized Riemannian Penrose inequality, we show that suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality is necessarily smooth in properly specified coordinates. If applied to hypersurfaces enclosing the horizon in a spatial Schwarzschild manifold, the result gives the rigidity of isometric hypersurfaces with the same mean curvature. This is a joint work with Pengzi Miao.
Feb 9, 2022 (Wed)
11:00AM
Zoom link		 Speaker: Sisi Shen (Columbia University)
Title: A Chern-Calabi flow on Hermitian manifolds

Abstract:
We discuss the existence problem of constant Chern scalar curvature metrics on a compact complex manifold and introduce a Hermitian analogue of the Calabi flow on compact complex manifolds with vanishing first Bott-Chern class.
Feb 16, 2022 (Wed)
10:00AM
Zoom link		 Speaker: Da Rong Cheng (University of Waterloo)
Title: Existence of constant mean curvature 2-spheres

Abstract:
This talk is based on joint work with Xin Zhou (Cornell). We show that in a 3-sphere equipped with an arbitrary Riemannian metric, for almost every H there exists a branched immersed 2-sphere with constant mean curvature H. Moreover, the existence extends to all values of H when the ambient metric has positive Ricci curvature.
Feb 23, 2022 (Wed)
10:00AM
Zoom link		 Speaker: Mikhail Karpukhin (Caltech)
Title: Optimization of Laplace and Steklov eigenvalues with applications to minimal surfaces

Abstract:
The study of optimal upper bounds for Laplace eigenvalues on closed surfaces is a classical problem of spectral geometry going back to J. Hersch, P. Li and S.-T. Yau. Its most fascinating feature is the connection to the theory of minimal surfaces in spheres. Optimization of Steklov eigenvalues is an analogous problem on surfaces with boundary. It was popularised by A. Fraser and R. Schoen, who discovered its connection to the theory of free boundary surfaces in Euclidean balls. Despite many widely-known empiric parallels, an explicit link between the two problems was discovered only in the last two years. In the present talk, we will show how Laplace eigenvalues can be recovered as certain limits of Steklov eigenvalues and discuss the applications of this construction to the geometry of minimal surfaces. The talk is based on joint works with D. Stern.
Mar 2, 2022 (Wed)
11:00AM
Zoom link		 Speaker: Bin Guo (Rutgers University)
Title: Uniform estimates for complex Monge-Ampere and fully nonlinear equations

Abstract:
Uniform estimates for complex Monge-Ampere equations have been extensively studied, ever since Yau’s resolution of the Calabi conjecture. Subsequent developments have led to many geometric applications to many other fields, but all relied on the pluripotential theory from complex analysis. In this talk, we will discuss a new PDE-based method of obtaining sharp uniform $C^0$ estimates for complex Monge-Ampere (MA) and other fully nonlinear PDEs, without the pluripotential theory. This new method extends more generally to other interesting geometric estimates for MA and Hessian equations. This is based on the joint works with D.H. Phong, F. Tong.
Mar 9, 2022 (Wed)
11:00AM
Zoom link		 Speaker: Yangyang Li (Princeton University)
Title: Minimal hypersurfaces in a generic 8-dimensional closed manifold

Abstract:
In the recent decade, the Almgren-Pitts min-max theory has advanced the existence theory of minimal surfaces in a closed Riemannian manifold $(M^{n+1}, g)$. When $2 \leq n+1 \leq 7$, many properties of these minimal hypersurfaces (geodesics), such as areas, Morse indices, multiplicities, and spatial distributions, have been well studied. However, in higher dimensions, singularities may occur in the constructed minimal hypersurfaces. This phenomenon invalidates many techniques helpful in the low dimensions to investigate these geometric objects. In this talk, I will discuss how to overcome the difficulty in a generic 8-dimensional closed manifold, utilizing various deformation arguments. En route to obtaining generic results, we prove the generic regularity of minimal hypersurfaces in dimension 8. This talk is partially based on joint works with Zhihan Wang.
Mar 16, 2022 (Wed)
11:00AM
Zoom link		 Speaker: Ved Datar (Indian Institute of Science)
Title: Some new rigidity results in complex geometry

Abstract:
I will discuss two new results on rigidity in Kähler geometry. I will first talk about the (almost) rigidity of Kähler manifolds with positive Ricci curvature and almost maximal volume. I will then discuss the rigidity of Kähler manifolds with positive bi-sectional curvature and maximal diameter. The first result is a complex analogue of a famous volume rigidity theorem of Colding, and the second result is the complex analogue of the well-known maximal diameter theorem of Cheng. This is joint work with Harish Seshadri and Jian Song.
Mar 23, 2022 (Wed)
5:00PM
Zoom link		 Speaker: Tobias Lamm (Karlsruhe Institute of Technology)
Title: Diffusive stability results for the harmonic map flow and related equations

Abstract:
The goal of this talk is to introduce the audience to the theory of diffusive stability in the context of the harmonic map flow. This theory is useful when studying stability results for parabolic equations and we will illustrate its use for geometric equations such as the harmonic map flow. Additionally, we use this theory in order improve various uniqueness results for solutions with rough initial data.
Mar 30, 2022 (Wed)
11:00AM
Zoom link		 Speaker: Teng Fei (Rutgers University)
Title: The Type IIA flow and its applications in symplectic geometry

Abstract:
The equations of flux compactifications of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. It can also be used to prove a stability result about Kahler structures. This is based on joint work with Phong, Picard and Zhang.
Apr 6, 2022 (Wed)
11:00AM
Zoom link		 Speaker: Yu Li (University of Science and Technology of China)
Title: Rigidity theorems of Ricci shrinkers

Abstract:
The Ricci shrinker plays an essential role in studying the singularities of the Ricci flow. We shall present some recent rigidity theorems in this talk, stating that some model spaces have no nearby Ricci shrinker in the Gromov-Hausdorff topology.
Apr 13, 2022 (Wed)
11:00AM
Zoom link		 Speaker: Antoine Song (University of California Berkeley)
Title: The spherical Plateau problem

Abstract:
For any closed oriented manifold with fundamental group G, or more generally any group homology class h for a group G, I will introduce an infinite codimension Plateau problem associated with h. This comes from an invariant called the spherical volume, which was defined by Besson-Courtois-Gallot in their study of the volume entropy. I will discuss some uniqueness and existence results for Plateau solutions.
Apr 20, 2022 (Wed)
11:00AM
Zoom link		 Speaker: Nicholas McCleerey (University of Michigan)
Title: Lelong Numbers of $m$-Subharmonic Functions Along Submanifolds

Abstract:
We study the possible singularities of an $m$-subharmonic function $\varphi$ along a complex submanifold $V$ of a compact Kahler manifold, finding a maximal rate of growth for $\varphi$ which depends only on $m$ and $k$, the codimension of $V$. When $k < m$, we show that $\varphi$ has at worst log poles along $V$, and that the strength of these poles is moreover constant along $V$. This can be thought of as an analogue of Siu's theorem. This is joint work with Jianchun Chu.





Past Seminars



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