Handbook of Workshop: Spectral Geometry

Transcription

1 Handbook of Workshop: Spectral Geometry December 22-23, 2016

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3 Workshop on Spectral Geometry General Information: The aim of the workshop is to bring together experts in spectral geometry to discuss recent progress on related topics. Workshop Location: Room 2201, Guanghua East Building (main), Speakers: Hua Chen, Xianzhe Dai, Xiaolong Han, Genqian Liu, Shiping Liu, Fanghua Lin, Zhiqin Lu, Fang Wang, Fengyu Wang, Wenjiao Yan, Jialin Zhu, Miaomiao Zhu, Wuhan University UC Santa Barbara California State University, Northridge Beijing Institute of Technology Courant Institute, New York University UC Irvine Shanghai Jiao Tong University Tianjin University Beijing Normal University Shanghai Center for Mathematical Sciences Shanghai Jiao Tong University

4 Scientific Committee: Jixiang Fu, Jiaxing Hong, Jiayu Li, Xinan Ma, Organizers: Bobo Hua: Zuoqin Wang: Sponsors: School of Mathematical Sciences, Shanghai Center for Mathematical Sciences, Workshop secretaries: Huihui Chen: Jiacheng Huang: Yan Huang:

5 Transportation: How to get to Fudan? There are three choices from Pudong Airport (PVG) to : Take taxi (about 50 minutes, about 170 RMB). Take Maglev Speed Train to Long Yang Road Station (50 RMB), then take taxi (about 60 RMB). Take Airport Bus Line 4, get off at Wu Jiao Chang Station (20 RMB). Then walk about 10 minutes. Take Shanghai metro line No. 2 to Guanglan Road, get off and take again No. 2 (the one to city center) to Nanjing East Road, transfer to No. 10, get off at Jiangwan Stadium Station. Then walk about 10 minutes. There are two choices from Hongqiao Airport (SHA) to : Take taxi (about 95 RMB). Take Shanghai metro line No. 10 to Jiangwan Stadium Station (5 RMB). Then walk about 10 minutes. From Shanghai Railway Station to : Exit at the North Exit, then take Bus No. 942 to. Or take taxi (about 40 RMB). From Shanghai South Railway Station to : Take Shanghai metro line No. 3 to DaBaiShu Station, then take taxi (about 14 RMB) or take Bus No. 133 to Fudan Universtiy. Or take taxi (about 95 RMB). Further information The following links point to pages that contain more information about Shanghai: Shanghai China Shanghai Travel Guide

6 Schedule: Thursday(12.22) Friday(12.23) 8:55 Opening 9:00-9:45 Hua Chen 9:00-9:45 Zhiqin Lu Morning Tea break (30 min) Tea break (30 min) 10:15-11:00 Fengyu Wang 10:15-11:00 Xiaolong Han 11:10-11:55 Fang Wang 11:10-11:55 Miaomiao Zhu 14:00-14:45 Xianzhe Dai 14:00-14:45 Genqian Liu Afternoon Tea break (30 min) 15:15-16:00 Shiping Liu Tea break (30 min) 15:15-16:00 Wenjiao Yan 16:10-16:55 Fanghua Lin 16:10-16:55 Jialin Zhu

7 Titles and Abstracts: 1. Hua Chen, Wuhan University Title:Estimates of eigenvalues for a class of bi-subelliptic operators Abstratct:Let $X = (X_1, X_2, \cdots, X_m)$ be a system of real smooth vector fields defined on a bounded open domain $\Omega\subset \mathbb{r}^n$ with smooth boundary $\partial\omega$ which is non-characteristic for $X$. If $X$ satisfies the Hormander's condition, then the vector fields is finitely degenerate and the sum of square operator $\triangle_{x}=\sum_{j=1}^{m}x_j^2$ is a subelliptic operator. In this talk, let $\lambda_{k}$ be the $k$-th Dirichlet eigenvalue for the bi-subelliptic operator $\triangle_{x}^2$ on $\Omega$, we shall use the subelliptic estimates to give the explicit lower bounds of $\lambda_{k}$, which is the extension of the classical result for bi-harmonic operator. 2. Xianzhe Dai, UC Santa Barbara Title: An eigenvalue problem for manifolds with conical singularities Abstract: Perelman's entropy functional plays an important role in analyzing Ricci flow. We develop Perelman's entropy functional for manifolds with conical singularity. It involves an eigenvalue problem for a singular Schrodinger operator and the analysis of the asymptotic behavior of its eigenfunctions near the conical singularity. This is joint work with Changliang Wang. 3. Xiaolong Han, California State University, Northridge Title: Small scale quantum ergodicity on negatively curved manifolds Abstract: Quantum ergodicity is the study of how the ergodicity (or chaos, i.e. with exponential instability) of a classical Hamiltonian system is reflected in its corresponding quantum system. For example, what implication does ergodicity (or chaos) of the geodesic flow on a compact Riemannian manifold have on the Laplacian eigenvalues and eigenfunctions? The quantum ergodic theorem states that, if the geodesic flow is ergodic, then a full density subsequence of the eigenfunctions tend equidistributed asymptotically. Small scale quantum ergodicity provides stronger and finer equidistribution than the quantum ergodic theorem. It has been applied to other estimates of eigenfunctions. I prove that quantum ergodicity up to a small scale holds on negatively cursed manifolds, using the fact that their geodesic flow display

8 exponential decay of correlation. I will also talk about the background on dynamical system, ergodicity, and chaos, in classical and quantum regimes, respectively. 4. Genqian Liu, Beijing Institute of Technology Title: Sharp higher-order Sobolev inequalities in the hyperbolic space Abstract: In this talk, we give the sharp k-th order Sobolev inequalities in the hyperbolic space for all positive integer k. This gives an answer to an open question raised by Aubin in [Aubin, Princeton University Press, Princeton (1982), pp ]. In addition, we prove that the associated Sobolev constants are optimal. 5. Shiping Liu, Title: Ricci curvature and eigenvalue estimates for discrete and continuous magnetic Laplacian Abstract: We present Lichnerowicz type estimates and (higher order) Buser type estimates for the magnetic Laplacian on discrete graphs with signatures and on closed Riemannian manifolds with magnetic potentials. These results relate eigenvalues, magnetic fields, Ricci curvature and Cheeger type constants. This is based on joint work with Michela Egidi, Florentin M\"unch, and Norbert Peyerimhoff. 6. Fanghua Lin, Courant Institute, New York University Title: Revisit Partition Problems for Dirichlet Eigenvalues Abstract: For a given bounded domain in $\mathbb{r}^n$, and a positive integer $m$, an optimal spectral partition of the domain is a partition of the domain into m none overlapping sub-domains in such a way that the sum of the first Dirichlet eigenvalues of these sum-domains being minimized. After reviewing some earlier results concerning the existence, regularity and the regularity of free interfaces of the optimal partitions, we shall discuss several difficult questions and conjectures. In particular, we shall prove that, as $m$ goes to infinite, the sum of the first Dirichlet eigenvalues of subdomains in an optimal partition, after a proper normalization, tend to a limiting value depending only on the dimension and the measure of the domain. 7. Zhiqin Lu, UC Irvine Title: On the ground state of quantum layers

9 Abstract: In this talk, we shall discuss the ground state of a quantum layer. Let $\Sigma$ be an embedded surface of $R^3$. A quantum layer is the set of points whose distance to $\Sigma$ is less than a fixed number $a$. We shall see that for many such quantum layers, the Dirichilet Laplacian has the ground state. 8. Fang Wang, Shanghai Jiao Tong University Title: Limit of sharp Sobolev inequalities Abstract: Consider the sharp Sobolev inequalities on the n-sphere. By assuming the dimension constant to be a continuous parameter, then the limit of sharp Sobolev inequalities gives the Moser-Trudinger inequality as n >2. However, this is a fake proof of the Moser-Trudinger since the dimension constants can only be integers. In this talk, I will mainly introduce a new point of view to make the limit to be mathematically true, by taking advantage of the fractional GJMS operators and their energy extension to the hyperbolic space. 9. Fengyu Wang, Tianjin University Title: Coupling by change of measure and applications Abstract: The classical Harnack inequality has been widely applied in PDEs and Geometry Analysis. However, as the inequality is essentially depend on the dimension of generator, it is invalid for infinite-dimensional models. To establish infinitedimensional Harnack inequalities, we introduce a new coupling method by change of measures, which allow the coupling to be successful at given fixed time. By looking at a simple stochastic differential equation, we explain the idea for applications of this method to the study of Harnack inequality, Bismut derivative formula and integration by parts formulas for Markov semigroups. 10. Wenjiao Yan, Beijing Normal University Title: Isoparametric foliation and the first eigenvalue problem Abstract: S. T. Yau conjectured that the first eigenvalue of every closed minimal hypersurface in the unit sphere is just its dimension. We prove this conjecture for minimal isoparametric hypersurfaces. As for the focal submanifolds, which are minimal submanifolds in the unit sphere, we show that their first eigenvalues are also equal to their dimensions in the non-stable range. Moreover, we give some estimate on the other eigenvalues.

10 11. Jialin Zhu, Shanghai Center for Mathematical Sciences Title: Scattering matrix and analytic torsion Abstract: For a compact manifold, which has a part isometric to a cylinder of finite length, we consider an adiabatic limit procedure, in which the length of the cylinder tends to infinity. We study the asymptotic of the spectrum of Hodge-Laplacian and the asymptotic of the L^{2}-metric on de Rham cohomology. As an application, we give a pure analytic proof of the gluing formula for analytic torsion. (joint work with Martin Puchol and Yeping Zhang) 12. Miaomiao Zhu, Shanghai Jiao Tong University Title: Regularity and blow-up analysis at the free boundary for approximate harmonic maps from surfaces Abstract: In this talk, we shall discuss the regularity and the blow-up analysis at the free boundary for approximate harmonic maps from surfaces with application to the two dimensional harmonicmap flow with free boundary. Also, we shall briefly discuss some related works. These are joint works with Jürgen Jost and Lei Liu.

11 Participants: Hua Chen Chuanqiang Chen Zhigang Chen Xianzhe Dai Feng Du Qi Ding Huijun Fan Jingwei Guo Xiaolong Han Yue He Qun He Xueping Huang Jiaxing Hong MinChun Hong Bobo Hua Yan Huang Jiacheng Huang Zhiqin Lu Genqian Liu Shiping Liu Fanghua Lin Xin Luo Hongquan Li Mijia Lai Yi Li Xianfu Liu Jing Mao Shengliang Pan Yuguo Qin Ke Shi Linlin Sun Weixu Su Yufei Shan Fengyu Wang Kui Wang Fang Wang Jiayong Wu Meng Wang Hao Wu Guoqiang Wu Wuhan University Zhejiang University of Technology Shantou University UC Santa Barbara Jingchu University of Technology Shanghai Center for Mathematical Sciences Peking University California State University,Northridge Nanjing Normal University Tongji University Nanjing University of Information Science & Technology The University of Queensland UC Irvein Beijing Institute of Technology Courant Institute, New York University Hunan University Shanghai Jiao Tong University Shanghai Jiao Tong University Wuhan University of Technology Tongji University Tongji University Tianjin University Suzhou University Shanghai Jiao Tong University Shanghai Maritime University Zhejiang University East China Normal University

12 Peng Wu Zuoqin Wang Weiwei Wang Dongmeng Xi Xiaowei Xu Jinju Xu Bo Xia Naqing Xie Ge Xiong Chengjie Yu Zhenwei Yu Songting Yin Wenjiao Yan Ling Yang Jialin Zhu Feifei Zhao Lingzhong Zeng Maomao Zhu Yanlong Zhang Chunqin Zhou Jiuru Zhou Shanghai Center for Mathematical Sciences Shanghai University Shanghai University Zhongshan University Tongji University Shantou University Shantou University Tongling University Beijing Normal University Shanghai Center for Mathematical Sciences Shantou University Jiangxi Normal University Shanghai Jiao Tong University Tongji University Shanghai Jiao Tong University Yangzhou University

13 Hotel: 1. Hotel: Fudan Yanyuan Hotel Address: No 270, Zhengtong Road (near Guoding Road), Yangpu district, Shanghai Tel: Hotel: Ramada Shanghai Wujiaochang Address: No 1888 Huangxing Road, Yangpu district, Shanghai Tel: Bus: From Ramada to Fudan campus Departure time: 8:20 in the morning

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