Geometric Analysis seminar talk list:
1.
Time: 14:00-15:00, Tuesday, January 9
Venue: E4-201
Speaker: Xingzhe Li, Cornell University
Title: Generic Scarring for Minimal Hypersurfaces in Manifolds Thick at Infinity with a Thin Foliation at Infinity
Abstract: In this talk, we present a generic scarring phenomenon for minimal hypersurfaces in a class of complete non-compact manifolds. In particular, for generic metrics on manifolds thick at infinity with a thin foliation at infinity, to each closed stable minimal hypersurface, there exists a sequence of closed minimal hypersurfaces, with area diverging to infinity, that accumulate along the stable hypersurface.
2.
Time: 15:30-16:30, Tuesday, January 9
Venue: E4-201
Speaker: Xuan Yao, Cornell University
Title: Applications of level set method in dimension 3
Abstract: In this talk, we introduce some recent developments in the applications of level set method in dimension 3, especially in the study of positive scalar curvature problems.
3.
Time: 10:30-11:30, Friday, January 19
Venue: E4-201
Speaker: Liding Huang, Xiamen University
Title: The stability of generalized Kahler Ricci Flow on toric Fano manifolds
Abstract: To construct canonical metrics and understand existence and moduli problems in generalzized Kahler geometry, Streets-Tian introduce the generalized Kahler-Ricci flow. We will discuss the stability of generalized Kahler Ricci Flow on toric Fano manifolds.
4.
Time: 9:00-10:30, Monday, March 4
Speaker: Jintian Zhu, Westlake University
Title: Positive scalar curvature metrics and aspherical summands
Abstract: In this talk, we consider obstruction for complete positive scalar curvature metrics on the aspherical summands $N\# X$ in dimensions up to five, where is $N$ 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.
Tencent Meeting: 760 6725 7412
Passcode: 678332
5.
Time: 9:00-11:00, Monday, March 11
Speaker: Xiaoshang Jin, Huazhong University of science and technology
Title: Willmore-type inequality for closed hypersurfaces in complete manifolds with Ricci curvature bounded below
Abstract: We establish a Willmore-type inequality for closed hypersurfaces in a complete Riemannian manifold of dimension $n+1$ with $Ric \geq -n g$ 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.
Tencent Meeting: 760 6725 7412
Passcode: 678332
6.
Time: 9:00-10:30, Monday, March 18
Speaker: Chao Li, NYU
Title: Stable minimal hypersurfaces in R^5
Abstract: 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.
ZOOM ID：952 802 9961
Password：314159
7.
Time: 13:00, Thursday, March 21
Venue: E4-201
Speaker: Zexuan Ouyang, Peking University
Title: Metric SYZ of Fermat type hypersurface
Abstract: In this talk, we investigate the behavior of the Calabi-Yau metrics of the Fermat family near the large complex structure limit. In particular, we will demonstrate that the SYZ conjecture holds in the Fermat case to some extent. This talk is based on the work of Yang Li.
8.
Time: 9:00, Friday, March 22
Speaker: Tong Freid, Havard University
Title: Monge Ampere equations and complete Calabi Yau metrics
Abstract: I will discuss a new free-boundary problem for a real Monge-Ampere equations that arises from the study of Calabi-Yau metrics. This is based on joint work with T. Collins and S.-T. Yau.
ZOOM ID：952 802 9961
Password：314159
9.
Time: 13:00, Friday, March 22
Venue: E4-201
Speaker: Song Yujian, Peking University
Title: Classification of gravitational instanton
Abstract: Gravitational instantons were introduced by Hawking in 1977. Mathematically, they are complete, non-compact hyperKahler 4-manifolds with curvature decay. It has been conjectured that there exist 4 main families of gravitational instantions according to their asymptotics by Cherkis and Kapustin. In 2022, this problem was solved by Sun and Zhang by using metric geometry and PDE analysis, resulting 6 different families (ALE,ALF, ALG, ALG*, ALH, ALH*). They study the collapsing limit of hyperKahler manifolds, and give an explicit description of the limit space. In particular, this calculates the asymptotic cones of gravitational instantons. In this talk, we will give a brief introduction about gravitational instantons. And then we discuss Sun–Zhang's work about the classification.
10.
Time: 9:00-11:00, Monday, March 25
Venue: E4-201
Speaker: Ruiming Liang, Peking University
Title: Asymptotically locally Euclidean metric with holonomy SU(m)
Abstract: In this talk, we first review the Eguchi-Hanson metric which is a typical example of ALE manifold. Generalizing the example, a calabi type conjecture for ALE manifolds can be stated. I'll explain the approach to address this problem following D.Joyce's paper.
11.
Time: 9:00-11:00, Tuesday, March 26
Venue: E4-201
Speaker: Linsheng Wang, Nanjing University
Title: Stable degenerations of klt singularities
Abstract: In this talk, I will introduce Xu-Zhuang's proof of finite generation result with respect to the associated ring of the normalized volume minimizer, which finishes the proof of stable degeneration conjecture of klt singularities.
12.
Time: 14:00-16:00, Wednesday, March 26
Venue: E4-201
Speaker: Shengxuan Zhou, Peking University
Title: Gromov-Hausdorff limits with large homology
Abstract: In this talk, we will introduce the Gromov-Hausdorff limit constructed by Hupp-Naber-Wang. In their construction, the limit space has dense topological singularities.
13.
Time: 9:00-11:00, Wednesday, March 27
Venue: E4-201
Speaker: Minghao Miao, Nanjing University
Title: On a Sufficient Criterion for the existence of cscK metrics
Abstract: In this talk, we will survey K. Zhang's work on a sufficient criterion for the existence of cscK metrics. We will discuss criteria for uniform J-stability as an intermediate step.
14.
Time: 14:00-15:00, Wednesday, March 27
Speaker: Bin Zhou, Peking University
Title: Regularity of variational problems with a convexity constraint
Abstract: We establish the interior $C^{1,\alpha}$ regularity of minimizers of a class of functionals with a convexity constraint, which includes the principal-agent problems studied by Figalli-Kim-McCann. The $C^{1,1}$ regularity was previously proved by Caffarelli-Lions in an unpublished note when the cost is quadratic, and recently extended to the case where the cost is uniformly convex with respect to a general preference function by McCann-Rankin-Zhang(arXiv:2303.04937). Our main result does not require the uniform convexity assumption on the cost function. In particular, we show that the solutions to the principal-agent problems with $q$-power cost are $C^{1,\frac{1}{q-1}}$ when $q > 2$ and $C^{1,1}$ when $1<q\leq 2$. Examples can show that this regularity is optimal when $q\geq 2$.
ZOOM ID：952 802 9961
Password：314159
15.
Time: 14:00-15:30, Thursday, March 28
Speaker: Haozhao Li, USTC
Title: On Ilmanen's multiplicity-one conjecture for mean curvature flow
Abstract: In this talk, we show that if the mean curvature of a closed smooth embedded mean curvature flow in 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 is at least of type-I. This is joint work with Bing Wang.
Tencent Meeting: 760 6725 7412
Passcode: 678332
16.
Time: 9:00-10:30, Monday, April 1
Speaker: Jingwen Chen, UPenn
Title: Mean curvature flow with multiplicity 2 convergence
Abstract: 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 $R^3$ 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.
ZOOM ID：952 802 9961
Password：314159
17.
Time: 9:00-10:30, Monday, April 8
Speaker: Man-Chun Lee, CUHK
Title: Gap theorem on manifold with pinched integral curvature bound
Abstract: 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.
Tencent Meeting: 760 6725 7412
Passcode: 678332
18.
Time: 15:30-16:30, Tuesday, April 9
Venue: E4-201
Speaker: Haobin Yu, Hangzhou Normal University
Title: Isoperimetry for asymptotically flat 3-manifolds with positive ADM mass
Abstract: In this talk, we will discuss the isoperimetry for asymptotically flat 3-manifolds with positive mass in large scale. We will show that for such manifolds each leaf of the canonical foliation is the unique isoperimetric surface for the volume it encloses. Our proof is based on "fill-in" argument and sharp isoperimetric inequality on asymptotically flat 3-manifold with nonnegative scalar curvature.
19.
Time: 15:30-16:30, Thursday, April 11
Venue: E4-201
Speaker: Mingyang Li, University of California，Berkeley
Title: Classification results for Hermitian non-Kahler gravitational instantons
Abstract: We will discuss some classification results for Hermitian non-Kähler gravitational instantons. There are three main results: (1) Non-existence of certain Hermitian non-Kähler ALE gravitational instantons. (2) Complete classification for Hermitian non-Kähler ALF/AF gravitational instantons. (3) Non-existence of Hermitian non-Kähler gravitational instantons under suitable curvature decay condition, when there is more collapsing at infinity (ALG, ALH, etc.). These are achieved by a thorough analysis of the collapsing geometry at infinity and compactifications.
20.
Time: 9:00-10:30, Monday, April 15
Speaker: Jinmin Wang, Texas A&M University
Title: Scalar curvature rigidity and Llraull's theorem
Abstract: 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.
Tencent Meeting: 760 6725 7412
Passcode: 678332
21.
Time: 14:00-15:00, Friday, April 19
Venue: E4-201
Speaker: Yalong Shi, Nanjing University
Title: Compactness of cscK metrics near the canonical class
Abstract: We shall prove that the set of csck metrics on minimal models constructed by Jian-Shi-Song is precompact with respect to the Gromov-Hausdorff topology. This is joint work with B. Guo, W. Jian and J. Song.
22.
Time: 9:00-10:30, Monday, April 22
Host: Jintian Zhu, ITS
Speaker: Yukai Sun, Peking University
Title: Positive mass theorem for asymptotically flat manifolds with isolated conical singularities
Abstract: 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.
Tencent Meeting: 760 6725 7412
Passcode: 678332
23.
Time: 14:00-15:00, Thursday, April 25
Venue: E4-201
Host: Tongrui Wang, ITS
Speaker: Zijun Wang, Shanghai Jiao Tong University
Title: Regularity for some geometric variational elliptic systems
Abstract: In this talk, we discuss the regularity issues of elliptic variational systems defined on manifolds. It places special emphasis on two aspects: the free boundary regularity of weakly H-surfaces into Riemannian manifolds and the interior regularity of weakly H-surfaces into static Lorentzian manifolds. These works are joint with Professor Miaomiao Zhu.
24.
Time: 15:30-16:30, Thursday, April 25
Venue: E4-201
Host: Tongrui Wang, ITS
Speaker: Rui Gao, Shanghai Jiao Tong University
Title: Recent results on bubbling analysis for approximate Harmonic maps and H-surfaces
Abstract: Compactness type results are crucial in the exploration of variational problems in both geometry and physics, as they provide a thorough understanding of the solutions’ behavior and the spaces they inhabit. A captivating example of this is the examination of the asymptotic behavior for sequences of approximate harmonic maps. This extends to a more general context, that is, for approximate surfaces with prescribed mean curvature, which are also called H-surfaces for simplicity. In this talk, we will discuss some recent progresses on the asymptotic and qualitative behavior of these entities.
25.
Time: 10:00-11:00, Friday, April 26
Venue: E4-201
Host: Xin Fu, ITS
Speaker: Ma Biao, BICMR
Title: On a fully nonlinear elliptic equation with differential forms
Abstract: We introduce a fully nonlinear PDE on Kahler manifolds with a differential form \Lambda. Such PDE unifies several important equations in complex geometry including Monge-Ampère equation, J-equation, and the deformed Hermitian Yang-Mills (dHYM) equation. Based on G.Chen's breakthrough on J-equation and dHYM equation, we prove analytical and algebraic criterions for the solvability of the equation, assuming certain positivity conditions on \Lambda. As an application of our results, we prove the conjecture of Collins-Jacob-Yau for the dHYM equation with small global phase. It is a joint work with Professor Hao Fang.
25.
Time: 10:00-11:00, Friday, April 26
Venue: E4-201
Host: Xin Fu, ITS
Speaker: Ma Biao, BICMR
Title: On a fully nonlinear elliptic equation with differential forms
Abstract: We introduce a fully nonlinear PDE on Kahler manifolds with a differential form \Lambda. Such PDE unifies several important equations in complex geometry including Monge-Ampère equation, J-equation, and the deformed Hermitian Yang-Mills (dHYM) equation. Based on G.Chen's breakthrough on J-equation and dHYM equation, we prove analytical and algebraic criterions for the solvability of the equation, assuming certain positivity conditions on \Lambda. As an application of our results, we prove the conjecture of Collins-Jacob-Yau for the dHYM equation with small global phase. It is a joint work with Professor Hao Fang.
26.
Time: 11:00-12:00, Friday, April 26
Venue: E4-201
Host: Tongrui Wang, ITS
Speaker: Mingxiang Li, Nanjing University
Title: On the positivity of the Q-curvatures of the conformal metrics
Abstract: In this talk, we will consider a conformal metric $g=u^{\frac{4}{n-2m}}|dx|^2$ on $\mathbb{R}^n$ with $n\geq 2m+1$. We show that if the higher order Q-curvature $Q^{(2m)}_g$ is positive with slow decay near infinity, the lower order Q-curvature $Q^{(2)}_g$ and $Q^{(4)}_g$ are both positive if $m$ is at least two. This talk is based on a joint work with Xingwang Xu.
27.
Time: 9:00-10:30, Monday, April 29
Host: Jintian Zhu, ITS
Speaker: Jingbo Wan, Columbia University
Title: Rigidity of Area Non-Increasing Maps
Abstract: 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.
ZOOM ID：952 802 9961
Password：314159
28.
Time: 15:30-17:00, Thursday, May 9
Venue: E4-201
Host: Xin Fu, ITS
Speaker: Junsheng Zhang，University of California，Irvine
Title: No semistability at infinity for Calabi-Yau metrics asymptotic to cones
Abstract: We proved a "no semistability at infinity" result for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in Donaldson-Sun's 2-step degeneration theory. As a consequence, we establish a polynomial convergence rate result and a classification result for complete Calabi-Yau manifolds with Euclidean volume growth and quadratic curvature decay. This is based on joint work with Song Sun.
