Peking-Westlake Geometric Analysis Seminar (Autumn 2024)

This is a Geometric Analysis Seminar held by

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

    • We welcome everyone who is interested in attending our seminar. If you would like to be added in the email list, please tell us by sending an email 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. We provide two choices of time for the convenience of the speaker (including a morning session and an afternoon session). The seminar will be arranged at

  • (Morning Session) 9:00-10:30 AM, Thursdays (Beijing time)
  • or (Afternoon Session) 2:00-3:30 PM, Tuesdays (Beijing time).

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

  • Tencent Meeting: 791 3541 8868 Password: 160273 .
  • Zoom: TBA (see below) .

    • The schedule is shown in the following table and you can find the details of the planned talks below this table.

    Week 1 (Sep 12, 9-10:30AM) Week 2 (Sep 19, 9-10:30AM) Week 3 (Sep 26, 9-10:30AM) Week 4 (National Holiday)
    Speaker: Shihang He (何世航) Speaker: Yihan Wang (王一涵) Speaker: Jinmin Wang (王晋民) (No arrangement)
    Week 5 (Oct 8/10) Week 6 (Oct 17, 9-10:30AM) Week 7 (Oct 24, 9-10:30AM) Week 8 (Oct 29, 2-3:30PM)
    (No arrangement) Speaker: Xiaoxiang Chai (柴小祥) Speaker: Guodong Wei (魏国栋) Speaker: Han Hong (洪寒)
    Week 9 (Nov 5, 2-3:30PM) Week 10 (Nov 14, 9-10:30AM) Week 11 (Nov 21, 9-10:30AM) Week 12 (Nov 28, 9-10:30AM)
    Speaker: Jie Zhou (周杰) Speaker: Yalong Shi (石亚龙) Speaker: Xuan Yao (姚萱) Speaker: Kai Xu (徐恺)
    Week 13 (Dec 5, 9-10:30AM) Week 14 (Dec 12, 9-10:30) Week 15 (Dec 17, 2-3:30PM) Week 16 (Dec 26, 9-10:30AM)
    Speaker: Zhihan Wang (王志涵) Speaker: Xingyu Zhu (朱星宇) Speaker: Zhichao Wang (王志超) Speaker: Jintian Zhu (朱锦天)

    Previous Seminars

  • Spring 2023
  • Autumn 2023
  • Spring 2024

    • Autumn 2024


    1. 2024 Sep 12, 9-10:30 AM,
    Tencent Meeting: 791 3541 8868, Password: 160273

    Speaker: Dr. Shihang He (Peking University)

    Title: Foliation of area minimizing hypersurfaces in asymptotically flat manifolds and Schoen's conjecture
    Abstract: In this talk, we present a foliation structure for asymptotically flat manifolds of dimension $4\leq n\leq 7$, each leaf being an area minimizing hypersurface. As an application, we prove a drift-to-infinity property of free boundary hypersurfaces in large cylinders lying in asymptotically flat manifolds with nonnegative scalar curvature and positive mass. This is joint work with Prof. Yuguang Shi and Prof. Haobin Yu.

    2. 2024 Sep 19, 9-10:30 AM,
    Tencent Meeting: 791 3541 8868, Password: 160273

    Speaker: Dr. Yihan Wang (Peking University)

    Title: Area-minimizing Hypersurfaces in Singular Ambient Manifolds
    Abstract: In this talk, I will report a recent work on the regularity of area-minimizing hypersurfaces in singular ambient manifolds whose tangent cones have nonnegative scalar curvature on their regular parts. The basic model is the non-existence of 2-dimensional minimizing hypercones in 3-dimensional ambient cones with isolated singularities and nonnegative curvatures. And then a codimension 3 regularity result can be obtained using a standard blow-up argument. Finally, I will introduce an example (Frank Morgan 2002) to show that this regularity result is sharp.

    3. 2024 Sep 26, 9-10:30 AM
    Tencent Meeting: 791 3541 8868, Password: 160273

    Speaker: Prof. Jinmin Wang (AMSS)

    Title: Sharp bottom spectrum and scalar curvature rigidity
    Abstract: In this talk, I will report our work on a sharp upper bound for the bottom spectrum of L2-Laplacian on complete manifolds with non-positive scalar curvature lower bound. A key ingredient of this result is is the application of higher index theory for Dirac operators. This talk is based on joint work with Bo Zhu.

    4. 2024 Oct 17, 9-10:30 AM
    Zoom: 912 9271 1520, Password: 517399

    Speaker: Prof. Xiaoxiang Chai (Postech University)

    Title: Scalar curvature rigidity of domains in a warped product
    Abstract: I will report my joint work with Wan Xueyuan and Gaoming Wang on scalar curvature rigidity of domains in a warped product. We found a condition analogous to the logarithmic concavity commonly seen in band width estimates and Llarull type theorems of of warped products. Then I will focus on the spinorial proof of a special case of Gromov dihedral rigidity conjecture in hyperbolic space. If time permits, I will my joint work with Wan on dihedral rigidity of polyhedral initial data sets.

    5. 2024 Oct 24, 9-10:30 AM
    Tencent Meeting: 791 3541 8868, Password: 160273

    Speaker: Prof. Guodong Wei (Sun Yat-sen University)

    Title: Volume growth and positive scalar curvature
    Abstract: In this talk, we shall talk about volume growth under the positive scalar condition. More precisely, for three dimensional complete, non-compact Riemannian manifolds with non-negative Ricci curvature and uniformly positive scalar curvature, we obtain the sharp linear volume growth ratio and the corresponding rigidity.

    6. 2024 Oct 29, 2-3:30 PM
    Tencent Meeting: 791 3541 8868, Password: 160273

    Speaker: Prof. Han Hong (Beijing Jiaotong University)

    Title: Rigidity and nonexistence of stable minimal hypersurfaces in $5$-dimensional manifolds
    Abstract: We prove that the nonnegative $3$-intermediate Ricci curvature and uniformly positive $k$-triRic curvature for certain range of $k$ implies the rigidity of complete noncompact two-sided stable minimal hypersurfaces in a Riemannian manifold $(X^5,g)$ with bounded geometry. The nonnegativity of $3$-intermediate Ricci curvature can be replaced by nonnegative Ricci and biRic curvature. In particular, there is no complete noncompact finite index minimal hypersurface in a closed $5$-dimensional manifold with positive sectional curvature. This extends some of results by Chodosh-Li-Stryker to $5$-dimensions. This is joint work with Zetian Yan.

    7. 2024 Nov 5, 2-3:30 PM
    Tencent Meeting: 791 3541 8868, Password: 160273

    Speaker: Prof. Jie Zhou (Capital Normal University)

    Title: Optimal volume bound and volume growth for Ricci-nonnegative manifolds with positive bi-Ricci curvature
    Abstract: In this presentation, we will discuss Gromov’s conjecture on the volume bound of Riemannian manifolds with nonnegative Ricci curvature and positive scalar curvature and its variant. As natural analogies, we care about the volume bound and volume growth of Ricci-nonnegative manifolds with positive bi-Ricci curvature and get the optimal bound. This is a joint work with Prof. Jintian Zhu from Westlake University.

    8. 2024 Nov 14, 9-10:30 AM
    Tencent Meeting: 791 3541 8868, Password: 160273

    Speaker: Prof. Yalong Shi (Nanjing University)

    Title: Green functions of GJMS operators on spheres, Gegenbauer polynomials and the rigidity problem
    Abstract: We derive explicit representation formulae of Green functions for GJMS operators on $S^n$ using Gegenbauer polynomials. These formulae have natural geometric interpretations concerning the extrinsic geometry. We shall also discuss the corresponding rigidity problem for the Green functions. A strong rigidity theorem in low dimensions is proven using the Positive Mass Theorem. This is joint work with Xuezhang Chen.

    9. 2024 Nov 21, 9-10:30 AM
    Zoom: 974 6747 0785, Password: 232215

    Speaker: Dr. Xuan Yao (Cornell University)

    Title: Scalar curvature comparison and rigidity
    Abstract: We introduce a new boundary comparison condition and prove a comparison-rigidity result which can be seen as a smooth analog of Gromov's conjecture on polihedron and a parallel of Shi-Tam's result.

    10. 2024 Nov 28, 9-10:30 AM
    Zoom: 974 6747 0785, Password: 232215

    Speaker: Dr. Kai Xu (Duke University)

    Title: Ricci lower bound in the spectral sense
    Abstract: We will discuss the geometry of manifolds whose first eigenvalue of $-\gamma\Delta+Ric$ is bounded from below. This is a global and weaker lower bound condition on Ricci curvature. We will talk about Bonnet-Myers and volume comparison theorems under such a condition, its sharp dependence on the parameter $\gamma$, as well as application to the stable Bernstein problem. The talk is based on my joint work with G. Antonelli.

    11. 2024 Dec 5, 9-10:30 AM
    Zoom: 974 6747 0785, Password: 232215

    Speaker: Dr. Zhihan Wang (Cornell University)

    Title: Generic Regularity of Minimal Submanifolds with Isolated Singularities
    Abstract: Singularities are commonly found in geometric variational objects, such as minimal submanifolds, where they are locally modeled on minimal cones. Despite the abundance of singularity models constructed in the literature, it is conjectured that in generic settings, they are significantly simpler. In this talk, we present a characterization of minimal cones that can serve as singularity models for minimal submanifolds with isolated singularities in a generic Riemannian manifold, without imposing additional constraint on dimension or codimension. As an application, we shall discuss a generic finiteness result of low area minimal hypersurfaces in nearly round 4-spheres. This is based on the joint work with Alessandro Carlotto and Yangyang Li.

    12. 2024 Dec 12, 9-10:30 AM
    Zoom: 974 6747 0785, Password: 232215

    Speaker: Xingyu Zhu (SLMath)

    Title: Fundamental groups and Ricci curvature
    Abstract: Understanding the topological obstruction of curvature lower bounds is a central theme in geometry. For complete manifolds of nonnegative Ricci curvature, it is previously known that when the manifold is compact its fundamental group is virtually abelian and when non-compact, any finitely generated subgroup of the fundamental group is virtually nilpotent. Moreover any torsion free nilpotent group can be realized a fundamental group of some manifold of nonnegative Ricci curvature. A natural question arising in this context is, how to recognize the virtually abelian fundamental group among the virtually nilpotent ones. In this talk we present a result that under extra linear volume growth condition, the fundamental group will always be virtually abelian and if one further assumes that the Ricci curvature is strictly positive, then the fundamental group is finite. We will highlight that, in the proof, the calculus on non-smooth metric measure spaces with lower Ricci curvature bounds (RCD spaces) plays a crucial role. This is joint work with Dimitri Navarro and Jiayin Pan.

    13. 2024 Dec 17, 2-3:30 PM
    Tencent Meeting: 791 3541 8868, Password: 160273

    Speaker: Prof. Zhichao Wang (Fudan University)

    Title: Scalar-mean rigidity theorem for compact manifolds with boundary
    Abstract: In this joint work with Bo Zhu and Jinmin Wang, we prove a scalar-mean rigidity theorem for compact Riemannian manifolds with boundary in dimension less than five by extending Schoen-Yau dimension reduction argument. As a corollary, we prove the sharp spherical radius rigidity theorem and best NNSC fill-in in terms of the mean curvature. Additionally, we prove a (Lipschitz) Listing type scalar-mean comparison rigidity theorem for these dimensions. Our results remove the spin assumption.

    14. Dec 26, 9-10:30 AM
    Tencent Meeting: 791 3541 8868, Password: 160273

    Speaker: Jintian Zhu (Westlake University)

    Title: Homological systole and bi-Ricci curvature
    Abstract: In this talk, we review basic notions and previous results on systolic inequalities, and then introduce our recent work estimating the homological systole of codimension one for closed Riemannian manifolds with positive bi-Ricci curvature.