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: 9:00-10:30, 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
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
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: 14:00-15:00, Tuesday, April 9
Venue: E4-201
Speaker: Shihang He, Peking University
Title: Twisted S^1 stability and positive scalar curvature obstruction on fiber bundles
Abstract: In 2006, Rosenberg made the S^1 stability conjecture, which states that for a compact manifold, the property of admitting no positive scalar curvature (PSC) metric is always preserved when multiplying S^1. In this talk, we will first review classical results and recent developments about PSC. Then we will investigate a twisted version of the above S^1 stability conjecture, as well as more generalized problem of the interaction between PSC obstruction on fiber and the total space of a fiber bundle.
19.
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.
20.
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.
21.
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
22.
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.
23.
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
24.
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.
25.
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.
26.
Time: 10:00-12:00, 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.
27.
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.
28.
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
29.
Time: 15:30-17:00, Thursday, May 9
Venue: E4-201
Host: Xin Fu, ITS
Speaker: Junsheng Zhang，University of California, Berkeley
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.
30.
Time: 9:00-10:30, Monday, May 13
Host: Jintian Zhu, ITS
Speaker: Yi Lai, Stanford University
Title: Riemannian and Kahler flying wing steady Ricci solitons
Abstract: 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\geq 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\geq 2, which answers a conjecture by H.-D.Cao in the negative. This is partly collaborated with Pak-Yeung Chan and Ronan Conlon.
ZOOM ID：952 802 9961
Password：314159
31.
Time: 15:00-16:30, Saturday, May 18
Venue: E4-233
Host: Jintian Zhu, ITS
Speaker: Jie Zhou, Capital Normal University
Title: Optimal rigidity estimates of varifolds almost minimizing the Willmore energy
Abstract: In this presentation, we talk about the stability of the Willmore functional. For an integral 2-varifold $V=\underline{v}(\Sigma,\theta)$ in $R^n$ with square integrable generalized mean curvature andfinite mass. If its Willmore energy is smaller thant $4\pi(1+\delta^2)$ and the mass is normalized to be $4\pi$, we show that $\Sigma$ is $W^{2,2}$ and bi-Lipschitz close to the round sphere in a quantitative way when $\delta<\delta_0\ll1$. For $n=3$, we show the sharp constant is $\delta_0^2=2\pi$. This is a joint work with Dr. Yuchen Bi.
32.
Time: 9:00-10:30, Monday, May 20
Host: Jintian Zhu, ITS
Speaker: Shihang He, Peking University
Title: Relative aspherical conjecture and higher codimensional obstruction to positive scalar curvature
Abstract: 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.
Tencent Meeting: 760 6725 7412
Passcode: 678332
33.
Time: 14:00-15:00, Tuesday, May 21
Venue: E4-201
Host: Jiyuan Han, ITS
Speaker: Linsheng Wang, Nanjing University
Title: Optimal destablizations of Fano varieties
Abstract: Delta invariant, which is also called of stability threshold, is an essential invariant in the study of K-stability of Fano varieties. In this talk, I will introduce Liu-Xu-Zhuang theory about the existence of divisorial valuations mininizing delta invariants.
34.
Time: 14:00-15:00, Wednesday, May 22
Venue: E4-201
Host: Jiyuan Han, ITS
Speaker: Yuto Yamamoto, RIKEN iTHEMS
Title: The Gross--Siebert program and non-archimedean SYZ fibrations
Abstract: For a maximally degenerate Calabi--Yau variety, the Berkovich retraction associated with a (good) minimal dlt model is regarded as an SYZ fibration in non-archimedean geometry. In general, the integral affine structure induced on the base space of the fibration differs from the one defined for the dual intersection complex of a toric degeneration in the Gross--Siebert program. In this talk, using tropical geometry, we construct non-archimedean SYZ fibrations whose bases are integral affine manifolds appearing in the Gross--Siebert program for Calabi--Yau complete intersections of Batyrev--Borisov.
35.
Time: 10:00-11:00, Thursday, May 23
Venue: E4-201
Host: Jiyuan Han, ITS
Speaker: Zexuan Ouyang, Peking University
Title: Constructing Special Lagrangian Fibrations via Higgs Bundles and Affine Structures
Abstract: In this talk, we explore the construction of special Lagrangian (SLag) fibrations using Higgs bundles, based on the work of Heller, Ouyang, and Pedit. We will discuss how solutions to Hitchin's equations provide an affine structure on the base space, derived from a hyperbolic affine sphere and a parabolic Higgs bundle. This affine structure is crucial for forming SLag fibrations, key to understanding mirror symmetry and Calabi-Yau manifolds.
36.
Time: 14:00-15:00, Friday, May 24
Venue: E4-201
Host: Jiyuan Han, ITS
Speaker: Yu Li, University of Science and Technology of China
Title: Uniqueness of the Tangent Flow of the Ricci Flow
Abstract: We prove the uniqueness of the tangent flow of the Ricci flow when one tangent flow is a generalized cylinder. The proof is based on a quantitative characterization of the rigidity of compact Ricci shrinkers, a rigidity inequality of mixed orders on generalized cylinders, and the method of contraction and extension developed by Colding and Minicozzi. This is joint work with Wenjia Zhang.
37.
Time: 14:00-15:00, Tuesday, May 28
Venue: E4-201
Host: Jiyuan Han, ITS
Speaker: Yueqing Feng, UC Berkeley
Title: A gluing construction of constant scalar curvature Kähler metrics of Poincaré type
Abstract: In this talk, we construct new examples of constant scalar curvature Kähler(cscK) metrics of Poincaré type from existing cscK ones. The construction is obtained via gluing a cscK metric on a compact Kähler manifold to a complete scalar-flat Kähler metric of Poincaré type on the complex space removing the origin. Assuming the compact Kähler manifold has no non-trivial holomorphic vector field, we prove the existence of cscK metrics of Poincaré type on this compact manifold removing finitely many points.
38.
Time: 14:00-15:00, Wednesday, May 29
Venue: E4-201
Host: Jiyuan Han, ITS
Speaker: Ziyi Zhao, Peking University
Title: Steady Ricci soliton with nonnegative curvature away from a compact set
Abstract: In this talk, we analysis the blow-down solutions for $n$-dimensional $(n\ge 4)$ noncompact $\kappa$-noncollapsed steady gradient Ricci solitons $(M, g)$ with $\rm{Rm}\geq 0$ and ${\rm Ric}>0$ away from a compact set of $M$. As one of main results, we classified the $(n-1)$-dimensional compact split limit ancient Ricci flows. Consequently, we prove that $(M,g)$ with $\rm{Rm}\geq 0$ must be isometric to the Bryant Ricci soliton up to scaling, if there exists a sequence of normally rescaled Ricci flows of $(M,g)$, which converges subsequently to a family of shrinking quotient cylinders. The later improves a previous result of Brendle. This is a joint work with Xiaohua Zhu.
39.
Time: 14:00-15:00, Friday, May 31
Venue: E4-201
Host: Jiyuan Han, ITS
Speaker: Minghao Miao, Nanjing University
Title: Optimal Degenerations of K-unstable Fano Threefolds
Abstract: In this talk, we will propose a question of how to explicitly determine the optimal degenerations of the K-unstable Fano manifolds as predicted by the Hamilton-Tian conjecture. We answer this question for a family of K-unstable Fano threefolds (No 2.23 in Mori-Mukai's list), which has discrete automorphism groups and the normalized Kahler-Ricci flow develops Type II singularity. Our approach is based on a new method to check weighted K-stability, which generalizes Abban-Zhuang's theory to give an estimate of the weighted delta invariant by dimension induction. Some speculative relations between the delta invariant and the H invariant will also be discussed. This is based on a joint work with Linsheng Wang.
40.
Time: 11:00-12:00, Friday, June 14
Venue: E4-201
Host: Jiyuan Han, ITS
Speaker: Shijin Zhang, Beihang University
Title: A quantitative second order estimate for p-harmonic functions in manifolds under curvature-dimension condition
Abstract: In this talk, first I will introduced some results about the gradient estimate of p-harmonic functions on Riemannian manifolds, including the results of Kotschwar-Ni, Wang-Zhang, Sung-Wang. Then I will introduce the results about the quantitative second-order Sobolev estimate of for positive p-harmonic functions in Riemannian manifolds under Ricci curvature bounded from below and also for positive weighted p-harmonic functions in weighted manifolds under the Bakry-Émery curvature-dimension condition. This is a joint work with Jiayin Liu and Yuan Zhou.
41.
Time: 16:00, Friday, July 12
Venue: E4-201
Host: Xin Fu, ITS
Speaker: Song Dai, Tianjin University
Title: Existence of harmonic metrics on nilpotent Higgs bundles over noncompact Riemann surfaces
Abstract: In this talk, we will first introduce the notions of Higgs bundles and harmonic metrics. Then we will survey some known results on the existence of harmonic metrics over noncompact Riemann surfaces. Our new result is that given a generically regular nilpotent harmonic bundle, there exists a (unique) maximal harmonic metric on the corresponding graded Higgs bundle. We will sketch the proof and show some applications. This is a joint work with Qiongling Li.
42.
Time: 10:00-11:00, Thursday, July 25
Venue: E4-201
Host: Jiyuan Han, ITS
Speaker: Yaxiong Liu, University of Maryland
Title: The eigenvalue problem of complex Hessian operators
Abstract: In a very recent pair of nice papers of Badiane and Zeriahi, they consider the eigenvalue problem of complex Monge-Ampere and complex Hessian, and show that the C^{1,\bar{1}}-regularity of eigenfunction for MA and C^alpha-regularity for complex Hessian. They posed a question about the C^{1,1}-regularity of the eigenfunction and the uniqueness. We give a positive answer and show the C^{1,1}-regularity and uniqueness of the eigenfunction. We also derive a number of applications, including a bifurcation-type theorem and geometric bounds for the eigenvalue. This is a joint work with Jianchun Chu and Nicholas McCleerey.
43.
Time: 14:00-15:00, Thursday, July 25
Venue: E4-201
Host: Jiyuan Han, ITS
Speaker: Yueqiao Wu, John Hopkins University
Title: K-semistability of log Fano conesingularities
Abstract: K-stability of log Fano cone singularities was introduced by Collins-Sz\' ekelyhidi to serve as a local analog of K-stability of Fano varieties. In the Fano case, the result of Li-Xu states that to test K-stability, it suffices to test the so-called special test configurations. In this talk, I will talk about a local version of this result for log Fano cones. Our method relies on a non-Archimedean characterization of local K-stability. This is joint work with Yuchen Liu.
44.
Time: 14:00-15:00, Thursday, August 1
Venue: E4-201
Host: Jiyuan Han, ITS
Speaker: Chi Li, Rutgers University
Title: Kahler compactification of C^n and minimal discrepancy of Fano cone singularities
Abstract: Let X be a smooth complex manifold. Assume that Y ⊂ X is a Kahler submanifold such that X \ Y is biholomorphic to C^n. We prove that (X, Y ) is biholomorphic to (P^n, P^{n−1}). We also study certain Kahler orbifold compactifications of C^n and, as an application, prove that on C^3 the flat metric is the only asymptotically conical Ricci-flat Kahler metric whose metric cone at infinity has a smooth link. As a key technical ingredient, a new formula for minimal discrepancy of isolated Fano cone singularities in terms of generalized Conley-Zehnder indices in symplectic geometry is derived.
45.
Time: 10:00-11:00, Wednesday, August 7
Venue: E4-201
Host: Jiyuan Han, ITS
Speaker: Gong Chen, Georgia Institute of Technology
Title: Recovery of the nonlinearity from the modified scattering map
Abstract: We consider the problem of recovering the nonlinearity in a nonlinear Schrödinger equation from scattering data, a problem for which there is a relatively large literature. We consider a new situation in which the equation does not admit standard scattering, but instead features the modified scattering behavior with logarithmic phase correction. We prove that even in this case, the modified scattering data suffices to determine the unknown nonlinearity. This is a joint work with J. Murphy (Oregon).
46.
Time: 9:00-10:30, Thursday, September 12
Tencent Meeting: 791 3541 8868
Host: Jintian Zhu, ITS
Speaker: 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, 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.
47.
Time: 9:00-10:30, Thursday, September 19
Tencent Meeting: 791 3541 8868
Host: Jintian Zhu, ITS
Speaker: 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.
48.
Time: 14:00-15:30, Tuesday, November 5
Tencent Meeting: 791 3541 8868
PASSCODE: 160273
Host: Jintian Zhu, ITS
Speaker: 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.
49.
Time: 9:00-11:00, Tuesday, November 12
Venue: E4-201
Host: Xin Fu, ITS
Speaker: Yaoting Gui, BICMR
Title: LIPSCHITZ REGULARITY OF HARMONIC MAPS FROM HEISENBERG GROUP INTO CAT(0) SPACE
Abstract: We prove the local Lipschitz continuity of energy minimizing harmonic maps between singular spaces, more specifically from the n-dimensional Heisenberg group into CAT(0) spaces. This is a joint work with Renan Assimos and Jürgen Jost.
50.
Time: 15:00-17:00, Wednesday, November 13
Venue: E4-201
Host: Jintian Zhu, ITS
Speaker: Zhu Ye, Capital Normal University
Title: Volume growth and asymptotic cones of manifolds with nonnegative Ricci curvature
Abstract: Let $M$ be an open (i.e. complete and noncompact) manifold with nonnegative Ricci curvature. We study the problem of whether the volume growth of $M$ is always greater than or equal to the dimension of some (or any) asymptotic cone of $M$. Assuming that $M$ is conic at infinity, we prove that there exists an asymptotic cone $Y$ of $M$ such that the upper box dimension of $Y$ is less than or equal to the infimum of volume growth order of $M$. In particular, if $M$ has nonnegative sectional curvature, then the Hausdorff dimension of its asymptotic cone is bounded above by the infimum of volume growth order of $M$. We also give an example of an open $n$-manifold $M$ with $\mathrm{sec}_M\geq0$, while the volume growth order of $M$ fluctuates between $1$ and $n$.
51.
Time: 9:00-10:30, Thursday, November 14
Tencent Meeting: 791 3541 8868
PASSCODE: 160273
Host: Jintian Zhu, ITS
Speaker: 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.
52.
Time: 9:00-10:30, Thursday, November 21
ZOOM: 974 6747 0785
PASSCODE: 232215
Host: Jintian Zhu, ITS
Speaker: 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.
53.
Time: 9:00-10:30, Thursday, November 28
ZOOM: 974 6747 0785
PASSCODE: 232215
Host: Jintian Zhu, ITS
Speaker: 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.
54.
Time: 9:30-10:30, Wednesday, December 4
Venue：E4-233
Host: Xin Fu, ITS
Speaker: Youmin Chen, Shantou University
Title: Quantization for biharmonic maps from non-collapsed degenerating Einstein 4-manifolds
Abstract: In this talk we shall consider the compactness problem for the moduli space of biharmonic maps from varying non-collapsed Einstein 4-manifolds to a fixed compact Riemannian manifold. Thanks to the well developed and beautiful compactness theory for non-collapsed Einstein 4-manifolds developed by Anderson, Tian, nakajima, bando, Kasue, Cheeger etc, we may study a sequence of biharmonic maps u_j : M_j → N from a sequence of non-collapsed closed Einstein 4-manifolds (M_j, g_j)with bounded Einstein constants, bounded diameters and bounded L^2 curvature energy to a compact Riemannian manifold (N, h) with uniformly bounded biharmonic energy. We establish a compactness theory modular finitely many bubbles.
55.
Time: 10:30-11:30, Wednesday, December 4
Venue：E4-233
Host: Xin Fu, ITS
Speaker: Rui Gao, Shanghai Jiao Tong University
Title: Existence of Prescribed Mean Curvature Surfaces of Abitrary Codimensions
Abstract: Constant Mean Curvature (CMC) and Prescribed Mean Curvature (PMC) surfaces are pivotal in diverse fields including mathematics, physics, and biology. They arise naturally in partitioning problems, isoperimetric problems, general relativity, two-phase interface problems, tissue growth etc. Despite the well-established existence theory for CMC and PMC hypersurfaces, constructing closed surfaces with prescribed mean curvature vector, admitting prescribed topology and controlled Morse index in general $n$-dimensional compact Riemannian manifold remains elusive. In this talk, we will outline our recent advancements in the existence theory for PMC spheres with arbitrary codimensions, contributing to a supplement of such area. This talk is based on the joint work with Prof. Miaomiao Zhu.
56.
Time: 9:00-10:30, Thursday, December 5
ZOOM: 974 6747 0785
PASSCODE: 232215
Host: Jintian Zhu, ITS
Speaker: Zhihan Wang, University of Chicago
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.
57.
Time: 9:00-10:30, Thursday, December 12
ZOOM: 974 6747 0785
PASSCODE: 232215
Host: Jintian Zhu, ITS
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.
58.
Time: 15:00, Friday, December 13
Venue: E4-201
Host: Xin Fu, ITS
Speaker: Rao Sheng, Wuhan University
Title: Uncountability incentive in deformation theory
Abstract: I will report mainly the progress on deformation limit and invariance of plurigenera of Moishezon manifolds over the unit disk. In particular, the role of uncountable distribution therein will be explained. If time permits, I will report one recent rigidity result of hyperbolic manifolds under the uncountable distribution condition. This talk is based on several joint works with Mu-Lin Li, I-Hsun Tsai, Kai Wang and Meng-jiao Wang.
