Peking-Westlake Geometric Analysis Seminar (Spring 2024)

This is a Geometric Analysis Seminar held by

  • Professor Yuguang Shi (Peking University)
  • and Dr. Jintian Zhu (Westlake University).

    • We welcome anyone who is interested in attending our seminar. If you want to be added in the email list, just contact us by sending emails to zhujintian AT westlake DOT edu DOT cn or haotz AT pku DOT edu DOT cn.

    Our seminar usually lasts 1.5 hours (including one-hour talk and half-hour discussion) but it would be flexible according to the speaker. The seminar will be arranged at (Beijing time)

  • 9:00-10:30 AM, Mondays
  • or 2:00-3:30 PM, Thursdays.

    • The talk will be given online with Tencent Meeting or Zoom (optional if the speaker cannot use Tencent Meeting).

  • Tencent Meeting: 760 6725 7412 Password: 678332.
  • Zoom: 952 802 9961 Password: 314159.

    • The whole schedule is collected in the following table and the detailed talk information is listed below the table.

    Week 1 (9-10:30AM, Feb 26) Week 2 (9-10:30AM, Mar 4) Week 3 (9-10:30AM, Mar 11) Week 4 (9-10:30AM, Mar 18)
    Speaker: Dr. Runzhang Wu (吴润璋) Speaker: Dr. Jintian Zhu (朱锦天) Speaker: Prof. Xiaoshang Jin (靳晓尚) Speaker: Prof. Chao Li (李超)
    Week 5 (2-3:30PM, Mar 28) Week 6 (9-10:30AM, Apr 1) Week 7 (9-10:30AM, Apr 8) Week 8 (9-10:30AM, Apr 15)
    Speaker: Prof. Haozhao Li (李皓昭) Speaker: Dr. Jingwen Chen (陈景文) Speaker: Prof. Man-Chun Lee (李文俊) Speaker: Dr. Jinmin Wang (王晋民)
    Week 9 (9-10:30AM, Apr 22) Week 10 (9-10:30AM, Apr 29) Week 11 (9-10:30AM, May 6) Week 12 (9-10:30AM, May 13)
    Speaker: Dr. Yukai Sun (孙昱凯) Speaker: Dr. Jingbo Wan (万静波) Speaker: Dr. Yifan Guo (郭逸帆) Speaker: Prof. Yi Lai (赖仪)
    Week 13 (9-10:30AM, May 20) Week 14 (9-10:30AM, May 27) Week 15 (9-10:30AM, Jun 3) Week 16 (9-10:30AM, Jun 10)
    Speaker: Dr. Shihang He (何世航) No arrangement Speaker: Dr. Zhu Ye (叶铸) Speaker: Prof. Martin Li (李文俊)
    Week 17 (9-10:30AM, Jun 17)
    Speaker: Dr. Jintian Zhu (朱锦天)

    Spring 2024


    1. 2024 Feb 26, 9-10:30 AM, Tencent Meeting

    Speaker: Runzhang Wu (Peking University)

    On open manifold admitting no complete positive scalar curvature metrics: What kind of manifolds admit no complete positive scalar curvature metrics is an important and interesting topic in Riemannian geometry. In this talk, we will generalize the Schoen-Yau-Shick (SYS) manifold to non-compact case, and give some examples of open SYS manifolds. We also discuss some topics about uniformly positive scalar curvature. This is joint with Prof. Yuguang Shi, Dr. Jintian Zhu and Dr. Jian Wang.

    2. 2024 Mar 4, 9-10:30 AM, Tencent Meeting

    Speaker: Dr. Jintian Zhu (Westlake University)

    Positive scalar curvature metrics and aspherical summands: In this talk, we consider obstruction for complete positive scalar curvature metrics on the aspherical summands $N\# X$ in dimensions up to five, where $N$ is a closed aspherical manifold and $X$ is an arbitrary non-compact manifold. We start by recalling related backgrounds as well as the slice-and-dice argument by Chodosh and Li, and then we show how to prove our main theorem with their techniques after introducing several essential improvements. This work is joint with Dr. Shuli Chen from Stanford University and Prof. Jianchun Chu from Peking University.

    3. 2024 Mar 11, 9-10:30 AM, Tencent Meeting

    Speaker: Prof. Xiaoshang Jin (Huazhong University of science and technology)

    Willmore-type inequality for closed hypersurfaces in complete manifolds with Ricci curvature bounded below: we establish a Willmore-type inequality for closed hypersurfaces in a complete Riemannian manifold of dimension $n+1$ with $Ric\geq -ng.$ It extends the classic result of Argostianiani, Fogagnolo and Mazzieri to the Riemannian manifold of negative curvature. As an application, we construct a Willmore-type inequality for closed hypersurfaces in hyperbolic space and obtain the characterization of geodesic sphere. This is a joint work with Jiabin Yin.

    4. 2024 Mar 18, 9-10:30 AM, Zoom

    Speaker: Prof. Chao Li (NYU)

    Stable minimal hypersurfaces in $R^5$: In this talk, I will explain why a complete stable minimal hypersurface in $R^5$ is flat. This is based on joint work with Chodosh, Minter and Stryker.

    5. 2024 Mar 28, 2-3:30 PM, Tencent Meeting

    Speaker: Prof. Haozhao Li (USTC)

    On Ilmanen's multiplicity-one conjecture for mean curvature flow: In this talk, we show that if the mean curvature of a closed smooth embedded mean curvature flow in $R^3$ is of type-I, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity one. This result confirms Ilmanen's multiplicity-one conjecture under the assumption that the mean curvature is of type-I. As a corollary, we show that the mean curvature at the first singular time of a closed smooth embedded mean curvature flow in $R^3$ is at least of type-I. This is joint work with Bing Wang.

    6. 2024 Apr 1, 9-10:30 AM, Zoom

    Speaker: Dr. Jingwen Chen (UPenn)

    Mean curvature flow with multiplicity $2$ convergence: Mean curvature flow (MCF) has been widely studied in recent decades, and higher multiplicity convergence is an important topic in the study of MCF. In this talk, we present two examples of immortal MCF in $\mathbb{R}^3$ and $S^n \times [-1,1]$, which converge to a plane and a sphere $S^n$ with multiplicity $2$, respectively. Additionally, we will compare our example with some recent developments on the multiplicity one conjecture and the min-max theory. This is based on joint work with Ao Sun.

    7. 2024 Apr 8, 9-10:30 AM, Tencent Meeting

    Speaker: Prof. Man-Chun Lee (CUHK)

    Gap theorem on manifold with pinched integral curvature bound : In Kahler geometry, Ni proved a optimal gap theorem on Kahler manifold with nonnegative bisectional curvature. In this talk, we will discuss some Riemannian analogy under nonnegative curvature and pinched integral curvature bound. This is based on joint work with Chan.

    8. 2024 Apr 15, 9-10:30 AM, Tencent Meeting

    Speaker: Dr. Jinmin Wang (Texas A&M University)

    Scalar curvature rigidity and Llraull's theorem: Llraull's theorem yields that one cannot increase the scalar curvature and the metric of the standard sphere simultaneously. Gromov conjectures this scalar curvature rigidity for incomplete metrics on spheres with two antipital points removed, and more generally warped product metrics. In this talk, I will present our proof of Gromov's conjecture under an extra condition using Dirac operator method, and a counterexample to Gromov's original statement. I will also give a brief introduction to the mu-bubble approach to this problem in dimension four. The talk is based on joint works with Simone Cecchini, Zhizhang Xie, and Bo Zhu.

    9. 2024 Apr 22, 9-10:30 AM, Tencent Meeting

    Speaker: Dr. Yukai Sun (Peking University)

    Positive mass theorem for asymptotically flat manifolds with isolated conical singularities : The well known Positive Mass Theorem states that for an asymptotically flat smooth manifold, if the scalar curvature is nonnegative, then the mass is also non negative. In this talk, we will discuss the Positive Mass Theorem for an asymptotically flat manifold with finitely isolated conical singularities.

    10. 2024 Apr 29, 9-10:30 AM, Zoom

    Speaker: Dr. Jingbo Wan (Columbia University)

    Rigidity of Area Non-Increasing Maps: In this talk, we discuss the approach of Mean Curvature Flow to demonstrate that area non-increasing maps between certain positively curved closed manifolds are rigid. Specifically, this implies that an area non-increasing self-map of $CP^n$, $n \geq 2$, is either homotopically trivial or is an isometry, answering a question by Tsai-Tsui-Wang. Moreover, by coupling the Mean Curvature Flow for the graph of a map with Ricci Flows for the domain and the target, we can also study the rigidity of area non-increasing maps from closed manifolds with positive 1-isotropic curvature (PIC1) to closed Einstein manifolds, where Prof. Brendle’s PIC1 Sphere Theorem is applied. The key to studying the rigidity of area non-increasing maps under various curvature conditions lies in the application of the Strong Maximum Principle along the MCF/MCF-RF. We will focus our attention on one particular case to illustrate the SMP argument. This is a joint work with Professor Man-Chun Lee and Professor Luen-Fai Tam from CUHK.

    11. 2024 May 6, 9-10:30 AM, Tencent Meeting

    Speaker: Dr. Yifan Guo (University of California, Irvine)

    Green's functions on minimal submanifolds: In this talk, we are going to discuss the following properties of Green's function on a minimal submanifold in $R^n$: lower and upper bounds, asymptotics near the pole and infinity as well as convergence under convergence of minimal submanifolds. We show that the Green's functions on area-minimizing boundaries have all of the mentioned properties. We also have $L^p$ convergence of the Green's functions for converging multiplicity 1 stationary varifolds. Outside of these classes of minimal submanifolds, the properties may fail and we discuss the reasons and some examples.

    12. 2024 May 13, 9-10:30 AM, Zoom

    Speaker: Prof. Yi Lai (Stanford University)

    Riemannian and Kahler flying wing steady Riccisolitons : Steady Ricci solitons are fundamental objects inthe study of Ricci flow, as they are self-similar solutions and often arise assingularity models. Classical examples of steady solitons are the most symmetric ones, such as the 2D cigar soliton, the $O(n)$-invariant Bryant solitons, and Cao’s $U(n)$-invariant Kahler steady solitons. Recently we constructed a family of flying wing steady solitons in any real dimension $n≥3$, which confirmed a conjecture by Hamilton in $n=3$. In dimension $3$, we showed allsteady gradient solitons are $O(2)$-symmetric. In the Kahler case, we also construct a family of Kahler flying wing steady gradient solitons with positive curvature for any complex dimension $n≥2$, which answers a conjecture by H.-D.Cao in the negative. This is partly collaborated with Pak-Yeung Chan and Ronan Conlon.

    13. 2024 May 20, 9-10:30 AM, Tencent Meeting

    Speaker: Dr. Shihang He (Peking University)

    Relative aspherical conjecture and higher codimensional obstruction to positive scalar curvature: Motivated by the solution of the aspherical conjecture up to dimension $5$ by Chodosh-Li and Gromov, we introduce a relative version of the aspherical conjecture. More precisely, we seek to explore the impact of a codimension $k$ submanifold $X$ on the existence of PSC (Positive Scalar Curvature) of the ambient space $Y$, under the relative aspherical condition that $\pi_i(Y,X) = 0, 2\leq i\leq k$. The formulation of the conjecture genralizes the aspherical conjecture and Rosenberg $S^1$ stability conjecture into a single framework, and is closely related to codim $2$ obstruction results by Hanke-Pape-Schick and Cecchini-Rade-Zeidler. In codim $3$ and $4$, we show how $3$-manifold obstructs the existence of PSC under our relative aspherical condition, the proof of which relies on a newly introduced geometric quantity called the spherical width. These results could be regarded as a relative version extension of the aspherical conjecture up to dim $5$.

    14. 2024 May 27, 9-10:30 AM, TBA

    Speaker: TBA

    TBA: TBA

    15. 2024 June 3, 9-10:30 AM, Tencent Meeting

    Speaker: Dr. Ye Zhu (Capital Normal University)

    Some rigidity theorems on open manifolds with nonnegative Ricci curvature: We will report some rigidity theorems on open manifolds with nonnegative Ricci curvature. The report is based on the recent work of Pan-Ye: Nonnegative Ricci curvature, splitting at infinity, and first Betti number rigidity, arXiv:2404.10145, and Ye: On manifolds with nonnegative Ricci curvature and the infimum of volume growth order<2, arXiv:2405.00852.

    16. 2024 June 10, 9-10:30 AM, Tencent Meeting

    Speaker: Prof. Martin Li (CUHK)

    Free boundary minimal surfaces via Allen-Cahn equation: It is well known that the semi-linear elliptic Allen-Cahn equation arising in phase transition theory is closely related to the theory of minimal surfaces. Earlier works of Modica and Sternberg et. al in the 1970’s studied minimizing solutions in the framework of De Giorgi’s Gamma-convergence theory. The more profound regularity theory for stationary and stable solutions were obtained by the deep work of Tonegawa and Wickramasekera, building upon the celebrated Schoen-Simon regularity theory for stable minimal hypersurfaces. This is recently used by Guaraco to develop a new approach to min-max constructions of minimal hypersurfaces via the Allen-Cahn equation. In this talk, we will discuss about the boundary behaviour for limit interfaces arising in the Allen-Cahn equation on bounded domains (or, more generally, on compact manifolds with boundary). In particular, we show that, under uniform energy bounds, any such limit interface is a free boundary minimal hypersurface in the generalised sense of varifolds. Moreover, we establish the up-to-the-boundary integer rectifiability of the limit varifold. If time permits, we will also discuss what we expect in the case of stable solutions. This is on-going joint work with Davide Parise (UCSD) and Lorenzo Sarnataro (Princeton). This work is substantially supported by research grants from Hong Kong Research Grants Council and National Science Foundation China.

    17. 2024 June 17, 9-10:30 AM, Tencent Meeting

    Speaker: Dr. Jintian Zhu (Westlake University)

    Llarull's theorem on punctured sphere with $L^\infty$ metric: The classical Llarull theorem states that a smooth metric on $n$-sphere cannot have scalar curvature no less than $n(n-1)$ and dominate the standard spherical metric at the same time unless it is the standard spherical metric. In this talk, we prove that Llarull's rigidity theorem holds for $L^{\infty}$ metrics on spheres with finitely many points punctured. This is a joint work with Prof. Jianchun Chu and Prof. Man-Chun Lee.