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
Dec 3, 2021
9:00AM
Speaker: Nick Edelen (University of Notre Dame)
Title: Degeneration of 7-dimensional minimal hypersurfaces with bounded index

Abstract: A 7D area-minimizing hypersurface $M$ can in general have a discrete singular set. The same is true if $M$ is only locally-stable for the area-functional, provided $H^6(sing M) = 0$. In this paper we show that if $M_i$ is a sequence of 7D minimal hypersurfaces with discrete singular set which are minimizing, stable, or have bounded index, and varifold-converge to some $M$, then the geometry, topology, and singular set of the $M_i$ can degenerate in only a very precise manner. We show that one can always ''parameterize'' a subsequence $i'$ by ambient, controlled bi-Lipschitz maps taking $\phi_{i'}(M_1) = M_{i'}$. As a consequence, we prove that the space of closed, $C^2$ embedded minimal hypersurfaces in a closed 8-manifold $(N, g)$ with a priori bounds $H^7(M) \leq \Lambda$ and $index(M) \leq I$ divides into finitely-many diffeomorphism types, and this finiteness continues to hold if one allows the metric $g$ to vary, or $M$ to be singular.

2021-22 Fall schedule

Date/Time/Venue Talks
Sep 24, 2021
9:00AM
Speaker: Brian Allen (University of Hartford)
Title: From $L^p$ to Metric Geometry Notions of Convergence For Riemannian Manifolds

Abstract: When studying geometric stability questions it is common to obtain $L^p$ estimates for a sequence of Riemannian manifolds as a first step. As a next step one often would like to improve from $L^p$ estimates to Sormani-Wenger Intrinsic Flat (SWIF) and/or Gromov-Hausdorff (GH) convergence. We will discuss joint work with Raquel Perales and Christina Sormani which allows one to do just that. We will also present a Morrey type inequality which characterizes the $p$ for which one should expect to obtain both GH and SWIF convergence versus when one should only expect SWIF convergence. An example will be given which shows that $L^p$ convergence is not well suited for metric geometry and we will give many examples which compare and contrast these notions that will provide intuition for all of the theorems discussed.
Oct 8, 2021
10:30AM
Speaker: Qi Ding (SCMS, Fudan University)
Title: Minimal hypersurfaces in manifolds with Ricci lower bound

Abstract: Let $N_i$ be a sequence of (n+1)-manifolds of Ricci curvature $\geq -n$ and the unit ball $B_1(p_i)$ in $N_i$ has volume $\geq v>0$. Suppose $B_1(p_i)$ converges to a metric ball $B_1(p_\infty)$ in the Gromov-Hausdorff sense. Let $M_i$ be a minimal hypersurface in $B_1(p_i)$ through $p_i$. Suppose the normalized volumes of $M_i$ are uniformly bounded.
In this talk, I will talk about the possible limits $M_\infty$ (of $M_i$) in $B_1(p_\infty)$ in the induced Hausdorff topology using Cheeger-Colding theory. One of main tools is the distance function from $M_\infty$. As an application, there is a Frankel property on cross sections of a class of metric cones, which is useful in proving certain Poincare inequality.
Oct 15, 2021
9:00AM
Speaker: Tin Yau Tsang (University of California, Irvine)
Title: On a spacetime positive mass theorem with corners

Abstract: Positive mass theorem asserts that an asymptotically flat initial data set satisfying the dominant energy condition has non-negative ADM mass. As revealed by the first minimal surface proof by Schoen and Yau, this theorem has a deep connection with scalar curvature geometry. Recently, Stern proposed a harmonic function's level set method to study scalar curvature. In this talk, we will talk about the applications of Stern's idea on positive mass theorem with corners which helps recall a relation between Physics (ADM mass, quasilocal mass) and geometry (Gromov’s questions on boundary geometry). This is partly a joint work with Hirsch (Duke) and Miao (Miami).
Oct 22, 2021
9:00AM
Speaker: Sebastien Picard (University of British Columbia)
Title: Non-Kahler Calabi-Yau conifold transitions

Abstract: A conifold transition is a process which connects two Calabi-Yau threefolds with different Hodge numbers. Such a transition may connect a Kahler Calabi-Yau threefold to a non-Kahler threefold. We will discuss solving the Yang-Mills equation on the tangent bundle through conifold transitions. This is joint work with T. Collins and S.-T. Yau.
Oct 29, 2021
9:00AM
Speaker: Christos Mantoulidis (Rice University)
Title: A nonlinear spectrum on closed manifolds

Abstract: The p-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace--Beltrami operator, which was defined by Gromov in the 1980s and corresponds to areas of a certain min-max sequence of hypersurfaces. By a recent theorem of Liokumovich--Marques--Neves, the p-widths obey a Weyl law, just like the eigenvalues do. However, even though eigenvalues are explicitly computable for many manifolds, there had previously not been any $\geq 2$-dimensional manifold for which all the p-widths are known. In recent joint work with Otis Chodosh, we found all p-widths on the round 2-sphere and thus the previously unknown Liokumovich--Marques--Neves Weyl law constant in dimension 2. Our work combines Lusternik--Schnirelmann theory, integrable PDE, and phase transition techniques.
Nov 5, 2021
3:00PM
Speaker: Shengwen Wang (University of Warwick)
Title: A Brakke type regularity for the parabolic Allen-Cahn equation

Abstract: We will talk about an analogue of the Brakke's local regularity theorem for the $\epsilon$ parabolic Allen-Cahn equation. In particular, we show uniform $C^{2,\alpha}$ regularity for the transition layers converging to smooth mean curvature flows as $\epsilon$ tend to 0 under the almost unit-density assumption. This can be viewed as a diffused version of the Brakke regularity for the limit mean curvature flow. This is joint work with Huy Nguyen.
Nov 12, 2021
9:00AM
Speaker: Jintian Zhu (BICMR, Peking University)
Title: Homotopical 2-systole inequalities for positive scalar curvature (PSC) manifolds

Abstract: In this talk, we will have a discussion on homotopical 2-systole inequalities for PSC manifolds. We plan to first review some recent progress on topology of PSC manifolds, and then turn to the homotopical 2-systole inequalities beyond the topology.
Nov 19, 2021
9:00AM
Speaker: Wangjian Jian (AMSS, Chinese Academy of Sciences)
Title: Diameter and Ricci curvature estimates for the collapsing long-time solution of the Kahler-Ricci flow

Abstract: We will first briefly recall the background of the Kahler-Ricci flow. Then we will introduce the set-up of collapsing long-time solution of the Kahler-Ricci flow, and recall some previous results. Next, we show how to reprove Tian-Zhang's relative volume comparison estimate by using recent progress by Bamler. Then we will show that this approach is also useful in long-time solution of the Kahler-Ricci flow without Ricci curvature bound assumption. Finally, if time permits, we will show how to apply iteration process to obtain the local Ricci bound if the Kodaira dim of the manifold is one.
Nov 26, 2021
9:00AM
Speaker: Pak Yeung Chan (University of California, San Diego)
Title: On a dichotomy of the curvature decay of steady Ricci soliton

Abstract: Ricci soliton arises naturally in the singularity analysis of the Ricci flow. Steady Ricci soliton is closely related to the Type II limit solution to the Ricci flow. There are two generic curvature decays for complete noncompact steady gradient Ricci soliton, namely linear and exponential decays. It is unclear if these are the only two possible decays. We show that this dichotomy holds for four dimensional complete noncompact non Ricci flat steady gradient Ricci soliton with at least linear curvature decay and proper potential function. A similar dichotomy is also shown in higher dimensions under the additional assumption that the Ricci curvature is nonnegative near infinity. As an application, we prove some classification results on steady soliton with fast curvature decay and obtain a dichotomy on the asymptotic geometry at spatial infinity. This talk is based on a joint work with Bo Zhu.
Dec 3, 2021
9:00AM
Speaker: Nick Edelen (University of Notre Dame)
Title: Degeneration of 7-dimensional minimal hypersurfaces with bounded index

Abstract: A 7D area-minimizing hypersurface $M$ can in general have a discrete singular set. The same is true if $M$ is only locally-stable for the area-functional, provided $H^6(sing M) = 0$. In this paper we show that if $M_i$ is a sequence of 7D minimal hypersurfaces with discrete singular set which are minimizing, stable, or have bounded index, and varifold-converge to some $M$, then the geometry, topology, and singular set of the $M_i$ can degenerate in only a very precise manner. We show that one can always ''parameterize'' a subsequence $i'$ by ambient, controlled bi-Lipschitz maps taking $\phi_{i'}(M_1) = M_{i'}$. As a consequence, we prove that the space of closed, $C^2$ embedded minimal hypersurfaces in a closed 8-manifold $(N, g)$ with a priori bounds $H^7(M) \leq \Lambda$ and $index(M) \leq I$ divides into finitely-many diffeomorphism types, and this finiteness continues to hold if one allows the metric $g$ to vary, or $M$ to be singular.

Past Seminars

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