Program
 Geometric Analysis Seminar Student No.： 80 Time： Sat 09:00-16:00, Apr.28/ Sun 09:00-11:30, Apr.29 Instructor： Xu Guoyi, Zhang Yingying Place： Conference Room 1, Jin Chun Yuan West Bldg. Starting Date： 2018-3-17 Ending Date： 2018-7-1

Geometric Analysis Workshop (May 19-20th)

2018-5-19
Speaker: Yong Lin (Renmin University)
Time: May 19th 10:00-10:45am
Location: Lecture Hall, Jin Chun Yuan West Building
Title: Trace formulas for discrete tori
Abstract:We will study the heat kernel and trace formulas on discrete tori. This is a joint work with A. Grigoryan and S.T. Yau.

10:45-11:00am: Break

Speaker: Haozhao Li (University of Science and Technology of China)
Time: May 19th 11:00-11:45am
Location: Lecture Hall, Jin Chun Yuan West Building
Title: On the multiplicity-one conjecture for mean curvature flow
Abstract: In this talk, I will talk about Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature. This is joint work with Bing Wang.

11:45am-2:00pm: Lunch

Speaker: Zuoqin Wang (University of Science and Technology of China)
Time: May 19th 2:00-2:45pm
Location: Lecture Hall, Jin Chun Yuan West Building
Title:On equivariant first eigenvalues for toric Kahler manifolds
Abstract:Given any compatible Kahler metric on a toric symplectic manifold, one can decompose the sequence of Laplacian eigenvalues into subsequences according to the weights of the torus action. We will study the properties of the first eigenvalues in each subsequence.

2:45-3:00pm: Break

Speaker: Mijia Lai (Shanghai Jiaotong University)
Time: May 19th 3:00-3:45pm
Location: Lecture Hall, Jin Chun Yuan West Building
Title: Obata type rigidity theorems for manifolds with boundary
Abstract:I will talk about some results on Obata type rigidity theorems for manifolds with boundary. Main applications are to characterize the equality case of first eigenvalue estimate with Robin boundary data and the equality case of some inequalities on Poincare-Einstein manifolds. This is joint with Xuezhang Chen and Fang Wang.

3:45-4:00pm: Break

Speaker: Wenshuai Jiang (Zhejiang University)
Time: May 19th 4:00-4:45pm:
Location: Lecture Hall, Jin Chun Yuan West Building
Title:The Structure of Noncollapsing Ricci Limit Spaces
Abstract:Let us consider a sequence of pointed n-manifolds (M^n_i,g_i,p_i) with uniform Ricci curvature lower bound Ric_i>=-(n-1)g_i and uniform volume lower bound Vol(B_1(p_i))>v>0. By Gromov's precompactness theorem, up to a subsequence (M^n_i,g_i,p_i) would converge in Gromov-Hausdorff sense to a metric space (X,d,p). In this talk we will discuss the structure of such metric space X and some applications. We will first introduce the results of Cheeger-Colding, and then discuss our recent improvement for the structure of the k-stratum of X. This is based on a joint work with Professor Jeff Cheeger and Aaron Naber.

6:00pm: Banquet

2018-5-20
Speaker: Hao Xu (Zhejiang University)
Time: May 20th 10:00-10:45am
Location: Lecture Hall, Jin Chun Yuan West Building
Title:TBD

10:45-11:00am: Break

Speaker: Xuezhang Chen (Nanjing University)
Time: May 20th 11:00-11:45am
Location: Lecture Hall, Jin Chun Yuan West Building
Title: The Han-Li conjecture in constant scalar curvature and constant boundary mean curvature problem
Abstract:We employ Mountain Pass Lemma to confirm the Han-Li conjecture for dimensions $3 \leq n \leq 7$ and present partial results of this conjecture in higher dimensions. Similarly as the boundary Yamabe problem, the toughest technical difficulty is to construct some test function satisfying the (PS) condition for a certain free functional. This is joint with Yuping Ruan (Nanjing Univ.) and Liming Sun (Johns Hopkins).

11:45am-2:00pm: Lunch

Speaker: Shicheng Xu (Capital Normal University)
Time: May 20th 2:00-2:45pm
Location: Lecture Hall, Jin Chun Yuan West Building
Title:Some progresses on a program of volume entropy and curvature bounds
Abstract:Several years ago we have started a program which concerns the interplay of the volume entropy and its behind geometry under lower curvature bounds. For a compact Riemannian manifold M, the volume entropy h(M) is defined to be the exponential growth rate of the volume of its universal cover at infinity. The study of volume entropy was first initiated many years ago (in 1980's ) with a tight relation from dynamic system. Motivated by recent works by Ledrappier-X.D. Wang and Kapovitch-Wilking around 2010, the main problems now we are interested in are: (1) if sectional curvature sec_M>=-1 and h(M)>n-2 where dim(M)=n, then is it true that M cannot be collapsed (i.e., volume vol(M)>v(n)>0)? Note that by Bishop volume comparison, h(M)<=n-1 which is the volume entropy of Hyperbolic n-manifolds; and we have proved that if h(M) is close enough to n-1 under lower Ricci curvature bound -(n-1) and diameter upper bound d, then M has to be diffeomorphic to a hyperbolic space and has a uniform positive lower volume bound. (2) what can be said if h(M) is close enough to 0 under bounded Ricci curvature and diameter d? It is conjectured that either h(M)>epsilon(n,d) or the fundamental group of M has a nilpotent group of finite index. We will talk about the latest progresses on this program. It is a joint work with Prof. Xiaochun Rong and Lina Chen.

2:45-3:00pm: Break

Speaker:Jian Ge (Peking University)
Time: May 20th 3:00-3:45pm
Location: Lecture Hall, Jin Chun Yuan West Building
Title: TBD

2018-4-29
Speaker: Bo Yang [Xiamen University]
Time: April 29th 09:00-10:00
Location: Conference Room 1, Jin Chun Yuan West Building
Title: On orthogonal complex structures (II)
Abstract:In the second talk, we present a result on the classification of orthogonal complex structures on real 6-dim tori.It is based on a joint work with Gabe Khan and Fangyang Zheng. One crucial ingredient is to study algebraic properties of Chern torsion tensor and apply a foliation technique. We will also revisit an interesting class of non-Kahler manifolds, previously studied by Blanchard, Calabi, Atiyah, Sommese, Catanese, and Borisov-Salamon-Viaclovsky.

Speaker: Martin Li(CUHK)
Time: April 29th 10:30-11:30
Location: Conference Room 1, Jin Chun Yuan West Building
Title: Free boundary minimal hypersurfaces II: existence and regularity
Abstract: In this talk, we will discuss some existence results for free boundary minimal hypersurfaces from three different perspectives: extremal Steklov eigenvalue problems, PDE and min-max theory. We will mention some open questions concerning the existence and uniqueness of free boundary minimal surfaces in the unit 3-ball. Some of these are joint work with N. Kapouleas, and X. Zhou.

2018-4-28
Speaker: Bo Yang [Xiamen University]
Time: April 28th 09:00-10:00
Location: Conference Room 1, Jin Chun Yuan West Building
Title: On orthogonal complex structures (I)
Abstract: A complex structure on a Riemannian manifold is called orthogonal if it is compatible with the Riemannian metric. In the first talk, we give an expository discussion on some important works on classification of orthogonal complex structures, for example, Lebrun's work on six-dim sphere and Gauduchon's work on real hyperbolic spaces.

Speaker: Xiangyu Liang [Beihang University]
Time: April 28th 10:30-11:30
Location: Conference Room 1, Jin Chun Yuan West Building
Title: Unique tangent behavior for 1-dimensional stationary varifold (I)
Abstracts: The uniqueness of tangent behavior is an important regularity property, and has been widely investigated in many circumstances in geometric measure theory and calculus of variations. In this talk we discuss the unique tangent behavior for stationary 1-varifolds in arbitrary Riemannian manifolds. Stationary varifolds are weak solutions for Plateaus problem in the setting of measures, defined as critical points of measure while deforming along any vector fields.We will  first introduce the background of the problem, following by definitions and examples. Then we will focus on tangent behavior for stationary varifolds, and discuss recent progresses on this subject.

Speaker: Martin Li [CHUK]
Time: April 28th 13:30-14:30
Location: Conference Room 1, Jin Chun Yuan West Building
Title: Free boundary minimal hypersurfaces I: compactness theorems
Abstract: In this talk, we will first define free boundary minimal hypersurfaces variational as critical points to the area functional. Then, we will describe several compactness results for certain classes of free boundary minimal hypersurfaces in Riemannian manifolds with boundary. In particular, we are interested in those which are either properly embedded or stable. Some of the results are joint work with Ailana Fraser, and Q. Guang - X. Zhou.

Speaker: Xiangyu Liang [Beihang University]
Time: April 28th 15:00-16:00
Location: Conference Room 1, Jin Chun Yuan West Building
Title: Unique tangent behavior for 1-dimensional stationary varifold (II)
Abstracts: The uniqueness of tangent behavior is an important regularity property, and has been widely investigated in many circumstances in geometric measure theory and calculus of variations. In this talk we discuss the unique tangent behavior for stationary 1-varifolds in arbitrary Riemannian manifolds. Stationary varifolds are weak solutions for Plateaus problem in the setting of measures, defined as critical points of measure while deforming along any vector fields.We will first introduce the background of the problem, following by definitions and examples. Then we will focus on tangent behavior for stationary varifolds, and discuss recent progresses on this subject.

2018-4-16

Speaker: Shouhei Honda [Tohoku University]
Time: April 16th 15:30-16:30

Location: Lecture Hall, Jin Chun Yuan West Building

Title: Local spectral convergence in metric measure spaces with Ricci bounds from below.
Abstract: In this talk we give a necessary and sufficient condition for local spectral convergence with respect to measured Gromov-Hausdorff convergence. By using this, we give an affirmative answer in a stronger form to a question on harmonic functions on Alexandrov spaces, raised by Anton Petrunin. This is a joint work with Luigi Ambrosio (Scuola Normale Superiore).

2018-4-12

Speaker: Shouhei Honda [Tohoku University]
Time: April 12nd 15:00-16:00

Location: Lecture Hall, Jin Chun Yuan West Building

Title: Weyl's law on metric measure spaces with Ricci bounds from below.
Abstract: In this talk we give a necessary and sufficient condition for Weyl's law on a metric measure space with Ricci bounds from below. This is a joint work with Luigi Ambrosio and David Tewodrose (Scuola Normale Superiore).

2018-3-22

Speaker: Renjie Feng [Peiking University]
Time: March 22nd 15:30-16:30

Location: Lecture Hall, Jin Chun Yuan West Bldg.
Title: Spectrum of SYK model
Abstract: The SYK model is a random matrix model arising from condensed matter theory in statistics physics and black hole theory in high energy physics.  In this talk, we will first review some elementary results in random matrix theory, then we will introduce the SYK model. I will explain the spectral properties of the random matrix of SYK model,  such as the global density where a phase transition is observed, the central limit theorem of the linear statistics and the concentration of measure theory. In particular, we will derive the large deviation principle when the number of interaction of fermions is 2.

2018-3-17&18

March 17th (Saturday)

Morning
9:30-10:30am: Reiko Miyaoka (Tohoku University)
10:30-10:45am: break
10:45-11:45am: Linfeng Zhou (East China Normal University)
11:45-1:30pm: Lunch

Afternoon
1:30-2:30pm: Reiko Miyaoka (Tohoku University)
2:30-2:45pm: break
2:45-3:45pm: Shicheng Xu (Capital Normal University)
3:45-4:00pm: break
4:00-5:00pm: Keita Kunikawa(Tohoku University)

March 18th (Sunday)

Morning
9:30-10:30am: Linfeng Zhou (East China Normal University)
10:30-10:45am: break
10:45-11:45am: Shicheng Xu (Capital Normal University)

Titles and Abstracts

Reiko Miyaoka (Tohoku University)
Title:  Exceptional values of  the Gauss map of complete minimal surfaces
Abstract:  Let M be a complete minimal surface in R^3. The Gauss map G  is a holomorphic map into CP^1=S^2.  The celebrated theorem of Fujimoto says that G omits at most 4 points of CP^1, and this result is sharp. On the other hand, M is called algebraic when its total curvature is finite. In this case, M is conformally a punctured Riemann surface, and the  so-called Weierstrass data is extendable beyond the punctures.  Osserman proved that the exceptional values are at most 3 for algebraic  minimal surfaces, and conjectured that it should be at most 2.  However, this problem is still open.

In my first talk, I introduce the minimal surface theory, and give many  examples with two exceptional values as well as how we construct them. In the second talk, introducing the Nevanlinna theory, we state a trial to  attack this conjecture by extending the Nevanlinna theory.

Linfeng Zhou (East China Normal University)
Title:  The isoperimetric problem in 2-dim Finsler space forms
Abstract:  In this talk, we will give an introduction of Finsler geometry and the isoperimetric problem in Finsler manifold.  By using the variational calculus theory, some recent joint work with Mengqing Zhan will be discussed.

Shicheng Xu (Capital Normal University)
Title:  Ricci curvature, diameter and Gap Vanishing Volume Entropy
Abstract:  In this talk we consider a conjecture about a gap phenomena on volume entropy: given n, d>0, there exists a constant \epsilon(n,d)>0 such that if a compact Riemannian n-manifold M satisfies that Ricci curvature >=-(n-1) and diameter <=d, then the volume entropy h(M)<\epsilon(n,d) implies that h(M)=0. It can be viewed as an quantitative version of a revised Milnor's problem:  whether a finitely presented group of non-exponential growth implies polynomial growth? We prove that a positive answer of Milnor's problem above implies the gap phenomena, and prove some partial results under some additional assumptions.

Keita Kunikawa(Tohoku University)
Title:  Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow
Abstract:  In this talk, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a Kahler- Einstein manifolds to more general Kahler manifolds including Fano manifolds by using the methodology proposed by T. Behrndt. We first consider a weighted measure on a Lagrangian manifold in such a Kahler manifold and investigate the variational problem of the Lagrangian for the weighted volume under Hamiltonian deformations. We call a stationary point and a local minimizer of the weighted volume f-minimal and Hamiltonian f-stable. We show such examples naturally appear in toric Fano manifolds. Moreover, we consider the generalized Lagrangian mean curvature flow which is introduced by Behrndt and also by Smoczyk-Wang. We generalize the result by H. Li, and show that if the initial Lagrangian is a small Hamiltonian deformation of an f-minimal and Hamiltonian f-stable Lagrangian, then the generalized MCF converges to an f-minimal one. This is a joint work with Toru Kajigaya.

2018-3-23

Speaker: Bernhard Hanke [University of Augsburg]

Time: March 23nd 11:00-12:00

Location: Lecture Hall, Jin Chun Yuan West Bldg.
Title: $\Gamma$-structures and symmetric spaces
Abstract:

$\Gamma$-structures are  weak forms of multiplications on closed oriented manifolds. As shown by Hopf the rational cohomology algebras of manifolds admitting $\Gamma$-structures are free over odd degree generators. We prove that this condition is also sufficient for the existence of $\Gamma$-structures on manifolds which are nilpotent in the sense of homotopy theory. This includes homogeneous spaces with connected isotropy groups.

Passing to a more geometric perspective we show that on compact oriented Riemannian symmetric spaces with connected isotropy groups and free rational cohomology algebras the canonical products  given  by geodesic symmetries define $\Gamma$-structures. This extends work of Albers, Frauenfelder and Solomon on $\Gamma$-structures on Lagrangian Grassmannians.