The seminar usually holds on Wednesday from 9:00-10:00 online. For more details, please visit here.
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Wednesday, February 19, 10:00-11:00 (Special time), Zoom link
(ID: 856 1651 6569, Code: 333115)
Man-Chun Lee (The Chinese University of Hong Kong) - Gap theorem on manifolds using Ricci flow - Abstract
In Kahler geometry, it was proved by Ni that curvature on nonflat Kahler manifold with nonnegative bisectional curvature cannot decay too fast at infinity. In this talk, we will discuss how Ricci flow can be used to prove the Riemannian gap theorem.
We will also discuss how the method related to existence of expanding soliton. This is based on joint work with P.-Y Chan.
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Wednesday, February 26, 16:30-17:30 (Special time), Zoom link
(ID: 830 0129 8209, Code: 397524)
Costante Bellettini (University College London) - PDE analysis on stable minimal hypersurfaces - Abstract
We consider properly immersed two-sided stable minimal hypersurfaces of dimension n. We illustrate the validity of curvature estimates for n \leq 6 (and associated Bernstein-type properties with an extrinsic area growth assumption). For n \geq 7 we illustrate sheeting results around "flat points".
The proofs rely on intrinsic (nonlinear) PDE analysis. The two main results extend respectively the analogous Schoen-Simon-Yau estimates (obtained for n \leq 5) and the Schoen-Simon sheeting theorem (valid for embeddings).
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Wednesday, March 5, 9:00-10:00, Zoom link
(ID: 838 2171 0038, Code: 506511)
Chuanjing Zhang (Ningbo University) - The L^p-approximate critical Hermitian structure on Higgs bundles - Abstract
In this talk, we consider the asymptotic behavior of the perturbed Hermitian-Yang-Mills equation, and prove the existence of L^p-approximate critical Hermitian structure on Higgs bundles. Then we give some applications. These works are joint with Chao Li, Shiyu Zhang and Professor Xi Zhang.
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Wednesday, March 12, 9:00-10:00, Zoom link
(ID: 897 1771 2616, Code: 909921)
Demetre Kazaras (Michigan State University) - Scalar curvature and codimension 2 collapse - Abstract
This talk is about the structure of Riemannian 3-manifolds satisfying a lower bound on their scalar curvature. These manifolds are models for spatial geometry in general relativity. Our motivational question will be "How flat is an isolated gravitational system with very little total mass?"
Objects like gravity wells and black holes can distort geometry without accumulating much mass, making this a subtle question. In addition to discussing progress, I will present a "drawstring" construction, which modifies a manifold near a given curve, reducing its length with negligible damage to a scalar curvature lower bound.
Unexpected examples are produced with relevance to a few problems. This construction extends ideas of Basilio-Dodziuk-Sormani and Lee-Naber-Neumayer, and is based on joint work with Kai Xu.
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Wednesday, March 19, 9:00-10:00, Zoom link
(ID: 861 8198 8793, Code: 608224)
Max Hallgren (Rutgers University) - Finite-Time Singularities of the Ricci Flow on Kähler Surfaces - Abstract
By work of Song-Weinkove, it is understood that the Ricci flow on any Kähler surface can canonically be continued through singularities in a continuous way until its volume collapses. This talk will discuss recent progress in understanding a more detailed picture of the singularity formation in this context.
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Wednesday, March 26, 16:00-17:00 (Special time), Zoom link
(ID: 881 0044 4833, Code: 926627)
Konstantinos Leskas (University of Athens) - Smooth approximations for CMC hypersurfaces with isolated singularities - Abstract
In this talk we consider a constant mean curvature (CMC) hypersurface in R^8, with an isolated singularity, that minimizes the prescribed mean curvature functional. We show that in a ball centered at the singularity there exist a sequence of smooth CMC hypersurfaces that converges to the initial one.
The proof relies on the Hardt-Simon foliation for area minimizing cones in R^8. The result extends the Hardt-Simon smooth approximation theorem, established for area minimizers.
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Wednesday, April 2, 9:00-10:00, Zoom link
(ID: 824 8060 6592, Code: 736980)
Chi Li (Rutgers University) - Two uniqueness results in Kahler geometry - Abstract
We will talk about two uniqueness results: (1) On C^3, any complete asymptotically conical Ricci-flat Kahler metric with a smooth link at infinity must be the flat metric. (2) Zoll manifolds of type CP^n with entire Grauert tubes are biholomorphic/isometric to CP^n with its Fubini-Study metric.
These two results are not directly related to each other but interestingly their proofs share parallel ideas and ingredients: (I) certain analytic compactification results, (II) connection between differential-geometric invariants (Conley-Zehnder index/Morse index) to certain algebra-geometric invariants (minimal log discrepancy/Fano index) and
(III) some classification results from algebraic geometry. We will also discuss how the two results are related to some classical open problems in complex geometry. This talk is based on joint works with Zhengyi Zhou and Kyobeom Song respectively.
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Wednesday, April 9, 16:00-17:00 (Special time), Zoom link
(ID: 845 0121 6281, Code: 315971)
Yang Li (Cambridge University) - Calabi-Yau metrics in the intermediate complex structure limit - Abstract
Calabi-Yau metrics can degenerate in a 1-parameter family by varying the complex structure, and a basic invariant is the dimension of the essential skeleton, which is an integer between 0 and n. The case of zero is the context of non-collapsed degeneration of Donaldson-Sun theory, while the case of n is the context of the SYZ conjecture.
We will discuss how to describe the Kahler potential at the C^0 level in the intermediate case for a large class of complete intersection examples.
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Wednesday, April 16, 9:00-10:00, Zoom link
(ID: 826 3876 4723, Code: 972908)
Chenzi Jin (University of Maryland) - Asymptotic analysis of stability thresholds - Abstract
Stability thresholds, particularly the \alpha_k and \delta_k invariants, are a fundamental topic in the theory of K-stability, with connections to various different fields such as algebraic geometry, convex geometry, and geometric analysis. In this talk, we investigate their asymptotic behavior, revealing new phenomena in both toric and non-toric settings.
In the toric setting, Ehrhart theory precisely describes the asymptotics via lattice point approximations of the moment polytope. We establish the stabilization of \alpha_k and derive an asymptotic expansion for \delta_k. In the general setting, we demonstrate that \alpha_k may fail to stabilize.
To study their asymptotics we analyze the Okounkov body and its discrete approximation, which are a generalization of moment polytopes on toric varieties. Using tools from convex geometry and lattice point enumeration techniques, we prove the first asymptotic result for \delta_k. Based on joint work with Y. Rubinstein and G. Tian.
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Wednesday, April 23, 9:00-10:00, Zoom link
(ID: 889 1206 8795, Code: 977611)
Davi Maximo (University of Pennsylvania) - Controlling Geometry and Topology in Dimensions 3 and 4 via Scalar Curvature - Abstract
In this talk, we will discuss recent applications of variational and parabolic methods to control the geometry and topology of manifolds with positive scalar curvature in dimensions three and four.
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Wednesday, April 30, 9:00-10:00, Zoom link
(ID: 870 5684 4637, Code: 490964)
Xiaoqi Huang (Louisiana State University) - Curvature and bounds for spectral projection operators on Riemannian manifolds - Abstract
We will first discuss the relation between curvature and L^q norm estimates of spectral projection operators on compact manifolds. In particular, we will demonstrate that the sign of the curvature of a compact space form can be inferred from the growth rate of L^q-norms of L^2-normalized quasimodes.
We will also discuss related recent work on the sharp spectral projection estimates on noncompact asymptotically hyperbolic surfaces with negative curvature, including all convex cocompact hyperbolic surfaces. This is based on joint works with Christopher Sogge, Zhongkai Tao and Zhexing Zhang.
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Wednesday, May 14, 9:00-10:00, Zoom link
(ID: 868 9207 1364, Code: 735547)
Yi Lai (University of California, Irvine) - 3-Manifolds with positive scalar curvature and bounded geometry - Abstract
We show that a complete contractible 3-manifold with positive scalar curvature and bounded geometry must be R^3. We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. This is joint work with O. Chodosh and K. Xu.
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Wednesday, May 21, 9:00-10:00, Zoom link
(ID: 826 3370 0574, Code: 676704)
Xiaoxiang Chai (Pohang University of Science and Technology) - Capillary mu-bubbles and tangent cone analysis in Llarull type theorems - Abstract
Despite being the weakest among notions of curvature, scalar curvature exhibits several rigidity properties, for instance, the positive mass theorem, the Geroch conjecture, and the more recent Gromov dihedral rigidity. In 1998, Llarull established a rigidity result for metrics on the round sphere under scalar curvature and metric comparison.
In this talk, we report on recent progress toward understanding scalar curvature rigid domains in warped product, where the warping function satisfies a log-concavity condition. Our main analytical tools are stable prescribed mean curvature surfaces with varying boundary contact angles, also known as stable capillary mu-bubbles.
Particular emphasis will be placed on the role of tangent cone analysis in the presence of point singularities on the boundary, where additional technical challenges arise.
This is joint work with Gaoming Wang (YMSC).
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Wednesday, May 28, 15:00-16:00 (Special time), Zoom link
(ID: 876 6541 2577, Code: 809687)
Thibaut Delcroix (Université de Montpellier) - Weighted Kähler geometry and semisimple principal fibrations - Abstract
In Kähler geometry, especially in the questions of existence of canonical Kähler metrics, the volume form \(\omega^n\) associated with a Kähler form \(\omega\) plays a central role.
In weighted Kähler geometry, we consider a Kähler manifold \(X\) equipped with a Hamiltonian action of a torus \(T\), a moment map \(\mu\) and associated moment polytope \(\Delta=\mu(X)\).
We fix a weight function \(v:\Delta \to (0,+\infty)\), then replace the volume form \(\omega^n\) by \(v\circ \mu \omega^n\). One can then define new canonical Kähler metrics: weighted solitons and weighted cscK metrics (introduced by Lahdili), which include most classical canonical Kähler metrics.
I will first introduce this weighted setting, then the (analytic) weighted delta invariant, a number that encodes the existence of weighted solitons. I will then present a sufficient condition of existence of weighted cscK metrics, in line with the J-flow approach of Song-Weinkove.
Then I will focus on the semisimple principal fibration cosntruction, a construction of varieties from a principal torus bundle and a fiber. The link with weighted Kähler geometry is that, under assumptions on the principal bundle, the Kähler geometry of the total space reduces to the weighted Kähler geometry of the fiber.
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Wednesday, June 4, 16:00-17:00 (Special time), Zoom link
(ID: 882 3895 8070, Code: 070076)
Henri Guenancia (Université de Bordeaux) - Kähler-Einstein metrics of negative curvature - Abstract
I will report on recent joint work with U. Hamenstädt where we show the existence in any complex dimension greater than or equal to two of compact Kähler-Einstein manifolds of negative curvature which are not uniformized by the unit ball.