The seminar usually holds on Wednesday from 9:00-10:00 online. For more details, please visit
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Wednesday, September 8, 15:00-16:00 (Special time), Zoom link
(ID: 872 2761 8837, Code: 157742)
Abdellah Lahdili (Beijing International Center for Mathematical Research) - Weighted K-stability and special metrics in Kahler and Sasaki geometry - Abstract
For a compact weighted extremal Kahler manifold we show the strict positivity of the weighted Donaldson-Futaki invariant of any non-product equivariant smooth Kahler test configuration with reduced central fibre,
a property also known as weighted K-stability on such test configurations. This provides a vast extension and a unification of a number of results concerning Kahler metrics satisfying special curvature conditions,
including constant scalar curvature Kahler metrics, extremal Kahler metrics, Kahler-Ricci solitons and their weighted extensions. For a class of fibre-bundles, we use the recent results of Chen-Cheng,
He in order to characterize the existence of extremal Kahler metrics, in terms of the coercivity of the weighted Mabuchi energy of the fibre. As an application, we establish a Yau-Tian-Donaldson correspondence
for weighted extremal Kahler metrics on sphere bundles over the product of cscK Hodge manifolds. This talk is based on joint work with Vestislav Apostolov and Simon Jubert.
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Wednesday, September 15, 9:00-10:00, Zoom link
(ID: 846 1313 1290, Code: 050126)
Chao Li (Courant Institute of Mathematical Sciences) - Stable minimal hypersurfaces in R^4 - Abstract
We prove that a complete, two-sided, stable minimal hypersurface in R^4 is flat. As consequences, we obtain curvature estimates for stable minimal hypersurfaces in Riemannian four-manifolds, and a structure theorem for complete,
two-sided minimal immersions in R^4 with finite Morse index.
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Wednesday, September 22, 9:00-10:00, Zoom link
(ID: 821 5664 6127, Code: 529643)
Siran Li (Shanghai Jiao Tong University) - Isometric immersions from geometric and PDE perspectives - Abstract
We report our recent work on a classical problem in geometric analysis: isometric immersions and/or embeddings of Riemannian and semi-Riemannian manifolds. The underlying PDE (partial differential equation) is the Gauss--Codazzi--Ricci equations.
Existence of isometric immersions is studied under various curvature conditions, via elliptic and hyperbolic PDE techniques. Weak continuity of isometric immersions is investigated with the help of the theory of compensated compactness.
Connections to other problems, including harmonic maps and Coloumb gauges, will also be discussed.
Our talk contains joint works with Gui-Qiang Chen and Marshall Slemrod.
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Wednesday, September 29, 9:00-10:00, Zoom link
(ID: 821 5664 6127, Code: 529643)
Jian Lu (South China Normal University) - Some recent results on the dual Orlicz-Minkowski problem - Abstract
The dual Orlicz-Minkowski problem arises from modern convex geometry. In the smooth case, it is equivalent to solving a class of Monge-Ampere type equations defined on the unit hypersphere.
These equations could be degenerate or singular in different conditions. We study some geometric flows related to the dual Orlicz-Minkowski problem. These flows involve Gauss curvature and functions of normal vectors and radial vectors.
By proving their long-time existence and convergence, we obtain new existence results of solutions to the dual Orlicz-Minkowski problem for smooth measures. We also discuss the uniqueness and nonuniqueness of solutions to a planar case of this problem.
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Wednesday, October 13, 9:00-10:00, Zoom link
(ID: 895 9574 4555, Code: 213167)
Yan Li (Beijing Institute of Technology) - Semistable limits of group compactifications - Abstract
Recently Han Jiyuan and Li Chi proved the algebraic uniqueness of the limit of Kahler-Ricci on a Fano manifold. In general, the limit space can be derived via two R-test configurations. In particular in the first one, the semistable degeneration,
it appears the soliton vector field. In this talk, we will first introduce a classification result of equivariant normal R-TC on group compactifications. Then we compute some algebraic invariants and find the semistable degeneration.
Finally for some special cases we will show that the semistable degeneration already provides the limit of the Kahler-Ricci flow.
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Wednesday, October 20, 9:00-10:00, Zoom link
(ID: 882 7465 0599, Code: 155336)
Qiang Guang (The Australian National University) - Existence of convex hypersurfaces with prescribed centro-affine curvature - Abstract
We introduce a functional involving weighted volume of convex bodies and their dual bodies and discuss the existence of critical points. When the ambient space is the sphere, finding critical criticals is equivalent to solving the Minkowski problem in the sphere (i.e., prescribing the Gauss curvature in the sphere).
When the ambient space is the Euclidean space, a critical point corresponds to a solution to the prescribed centro-affine curvature problem. By using Gauss curvature flow and a variational method, we discuss the existence of origin-symmetric solutions.
A crucial ingredient in our proof is a topological argument based on the study of a topological space of ellipsoids.
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Wednesday, October 27, 9:00-10:00, Zoom link
(ID: 844 1769 1567, Code: 002200)
Xin Fu (University of California, Irvine) - Kahler-Einstein metric near an isolated log canonical singularity - Abstract
We construct Kahler-Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable. In complex
dimension 2, we show that any complete Kahler-Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to one of the model metrics constructed by Kobayashi and Nakamura arising from hyperbolic geometry. If time
permits, optimal asymptotics on hyperbolic cusps will also be discussed. This talk is based on joint work with Ved Datar, Hans-Joachim Hein, Jiang Xumin and Jian Song.
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Wednesday, November 3, 9:00-10:00, Zoom link
(ID: 883 6219 0680, Code: 118143)
Ling Xiao (University of Connecticut) - Entire spacelike hypersurfaces with constant curvature in Minkowski space - Abstract
We prove that, in the Minkowski space, if a spacelike, (n-1)-convex hypersurface M with constant $\sigma_{n-1}$ curvature has bounded principal curvatures, then M is convex. Moreover, if M is not strictly convex, after an R^{n,1} rigid motion, M splits as a product $M^{n-1}\times R$.
We also construct nontrivial examples of strictly convex, spacelike hypersurfaces M with constant $\sigma_k$ curvature and bounded principal curvatures. This is a joint work with Changyu Ren and Zhizhang Wang.
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Wednesday, November 10, 9:00-10:00, Zoom link
(ID: 890 0054 5464, Code: 627492)
Siyuan Lu (McMaster University) - Mobius invariant equations in dimension two - Abstract
Conformally invariant equations in n\geq3 have played an important role in the study of \sigma_k-Yamabe problem in geometric analysis. In this talk, we will discuss a class of Mobius invariant equations in dimension two and then present a Liouville type theorem for such equations.
We will then discuss the \sigma_2-Nirenberg problem on \mathbb{S}^2. This is based on joint works with Yanyan Li and Han Lu.
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Wednesday, November 17, 16:00-17:00 (Special time), Zoom link
(ID: 864 4250 4459, Code: 070281)
Hoang-Chinh Lu (Université Paris-Saclay) - Complex Monge-Ampere equations on compact complex manifolds - Abstract
On a compact Hermitian manifold X we consider the complex Monge-Ampere equation with right-hand side f in L^p, p>1 and semipositive and big reference form omgea. We prove that there is a continuous solution which is smooth in a Zariski open set if an additional regularity assumption on the density f is assumed.
As an application, we obtain a singular Hermitian analogue of Yau's solution to the Calabi conjecture. This is a joint work with Vincent Guedj announced on arXiv:2107.01938.
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Wednesday, November 24, 9:00-10:00, Zoom link
(ID: 838 6740 7785, Code: 344668)
Zhichao Wang (University of British Columbia) - Min-max minimal hypersurfaces with multiplicity two - Abstract
Recently, X. Zhou proved the multiplicity one theorem for bumpy metrics in the Almgren-Pitts theory. This made it tempting to conjecture that for any metric there always exists a min-max varifold of multiplicity one.
However, in this talk, we will disprove this naive conjecture by constructing the first set of nontrivial and non-bumpy examples, where the varifold associated with a two-parameter min-max construction must have multiplicity two. This is a joint work with X. Zhou.
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Wednesday, December 1, 9:00-10:00, Zoom link
(ID: 884 0327 2402, Code: 600144)
Guohuan Qiu (Chinese Academy of Sciences) - Interior Hessian estimate for sigma_2 equations - Abstract
Motivated by isometric embedding problems, E.Heinz proved interior C^2 estimate for 2-d Monge-Ampere equations.
In this talk, I will introduce a new pointwise approach to the 2-d Monge-Ampere equation.
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Wednesday, December 8, 9:00-10:00, Zoom link
(ID: 841 8772 1220, Code: 430726)
Ruobing Zhang (Princeton University) - Collapsing geometry of Hyperkaehler four-manifolds - Abstract
This talk focuses on the recent resolution of the following three well-known conjectures in the study of Ricci-flat four manifolds (joint with Song Sun).
(1) Any volume collapsed limit of unit-diameter hyperkaehler metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3D torus by an involution, a singular special Kaehler metric on the topological 2-sphere, or the unit interval.
(2) Any complete non-compact hyperkaehler 4-manifold with quadratically integrable curvature, namely gravitational instanton, must have one of the following asymptotic model geometries: ALE, ALF, ALG, ALH, ALG* and ALH*.
(3) Any gravitational instanton can be compactified to an open dense subset of certain compact algebraic surface.
With the above classification results, we obtain a rather complete picture of the collapsing geometry of hyperkaehler four manifolds, i.e., classifications of Gromov-Hausdorff limit geometries, tangent cones, singularity formations, as well as all the rescaling bubbles.
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Wednesday, December 15, 9:00-10:00, Zoom link
(ID: 814 2281 1215, Code: 464145)
Bin Guo (Rutgers University, Newark) - 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 and C. Wang.
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Wednesday, December 22, 9:00-10:00, Zoom link
(ID: 820 0614 2218, Code: 366307)
Xiaolan Nie (Zhejiang Normal University) - Schwarz lemma: the case of equality and an extension - Abstract
Schwarz lemma is a fundamental tool in complex analysis and geometry. In this talk, we will discuss the equality case of the Schwarz lemmas of Yau, Royden and Tosatti. Then we will talk about an extension of Schwarz lemma to almost Hermitian setting in terms of holomorphic sectional curvatures.
This talk is based on a joint work with Haojie Chen.
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Wednesday, December 29, 9:00-10:00, Zoom link
(ID: 847 2354 5893, Code: 827858)
Ping Li (Tongji University) - The Hirzebruch genera, symmetric functions and multiple zeta values - Abstract
In this talk we first review several classical oriented and complex genera, and then explain how to derive the coefficients in front of the characteristic numbers, where the symmetric function theory and mutiple-(star) zeta values are involved.
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Wednesday, January 5, 9:00-10:00, Zoom link
(ID: 830 9599 2335, Code: 413864)
Jintian Zhu (Beijing International Center for Mathematical Research) - Area and Gauss-Bonnet inequalities with scalar curvature - Abstract
Understanding the largeness of positive scalar curvature manifolds is one of the central topics in the research on scalar curvature. In this talk, I will review some history and introduce area and Gauss-Bonnet inequalities for positive scalar curvature n-manifolds with (n-2)-directions large.
This talk is based on my recent work joint with M. Gromov.