**Young Mathematician Workshop on Differential and Metric Geometry**

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**Date: Aug 12 - 13, 2017**

**Place: Second Academic Building, room 827, Capital Normal University**

In order to disseminate current research developments in differential and metric geometry and to promote cooperation between young mathematicians in China, a Young Mathematician Workshop will be held in the School of Mathematical Sciences at Capital Normal University during August 12 - 13, 2017.

This workshop is sponsored by a research fund from Capital Normal University. It is planned to be held annually in the future.

**Academic Committee:**

Fuquan Fang (Capital Normal University)

Xiaochun Rong (Capital Normal University)

Zuhuan Yu (Capital Normal University)

Zhenlei Zhang (Capital Normal University)

**Organizers:**

Yanyan Niu (Capital Normal University)

Shicheng Xu (Capital Normal University)

Zuhuan Yu (Capital Normal University)

Zhenlei Zhang (Capital Normal University)

**Participants:**

Qingsong Cai, 蔡青松, Tianjin University of Technology

Daguang Chen, 陈大广, Tsinghua University

Li Chen, 陈立, Hubei University

Xiaoyang Chen (speaker), 陈小杨, Tongji University

Zhengxiang Chen, 陈正祥, Capital Normal University

Bo Dai, 戴波, Peking University

Xiaojuan Duan, 段孝娟, Xiamen University of Technology

James Dibble (speaker), the University of Iowa

Huabin Ge, 葛化彬, Beijing Jiaotong University

Jian Ge (speaker), 葛剑, Peking University

Jianquan Ge, 葛建全, Beijing Normal University

Juanru Gu (speaker), 顾娟如, Zhejiang University of Technology

Hongxin Guo, 郭洪欣, Wenzhou University

Shaochuang Huang, 黄少创, Yau Mathematical Sciences Center

Xiantao Huang (speaker), 黄显涛, Sun Yat-Sen University

Lingling Kong, 孔令令, Northeast Normal University

Gang Li, 李刚, Shandong University

Yong Luo, 罗勇, Wuhan University

Jing Mao (speaker), 毛井, Hubei University

Jiayin Pan (speaker), 潘佳垠, Rutgers University

Sheng Rao (speaker), 饶胜, Wuhan University

Hongliang Shao, 邵红亮, Chongqing University

Xiaole Su, 苏效乐, Beijing Normal University

Yanhui Su, 苏延辉, Fuzhou University

Zhongyang Sun, 孙忠洋, Huaibei Normal University

Jianming Wan, 万建明, Northwest University

Fang Wang (speaker), 王芳, Shanghai Jiaotong University

Guoqiang Wu, 吴国强, Zhejiang Sci-Tech University

Yunhui Wu, 吴云辉, Yau Mathematical Sciences Center

Guoyi Xu (speaker), 徐国义, Yau Mathematical Sciences Center

Haifeng Xu, 徐海峰, Yangzhou University

Xu Xu, 徐旭, Wuhan University

Xiaowei Xu, 许小卫, University of Science and Technology of China

Ling Yang, 杨翎, Fudan University

Chengjie Yu (speaker), 余成杰, Shantou University

Lingzhong Zeng, 曾令忠, Jiangxi Normal University

Liyou Zhang, 张利友, Capital Normal University

Huichun Zhang (speaker), 张会春, Sun Yat-Sen University

Yingying Zhang, 张蓥莹, Yau Mathematical Sciences Center

Yongsheng Zhang (speaker), 张永胜, Northeast Normal University

Hengyu Zhou, 周恒宇, Sun Yat-Sen University

**Titles and Abstracts**

**1. Xiaoyang Chen ****陈小杨****, Tongji University**

**Title: **Symplectic aspects of polar actions

**Abstract:** Polar actions are a special class of isometric group actions on Riemannian manifolds. We will give a symplectic look at polar actions and discuss its application in symplectic geometry. This is a joint work with Jianyu Ou.

**2.** **James Dibble, the University of Iowa**

**Title:** Totally geodesic maps into manifolds with no focal points

**Abstract:** Eells-Sampson and Hartman proved that the set of harmonic maps in each homotopy class of maps between compact Riemannian manifolds, where the domain has non-negative Ricci curvature and the target non-positive sectional curvature, is non-empty, path-connected, and equal to the set of totally geodesic maps in that class. It will be shown that this generalizes to energy-minimizing maps into manifolds with no focal points. These are manifolds whose universal covers satisfy a synthetic condition: Each point and maximal geodesic are connected by a unique geodesic that intersects the latter perpendicularly. The proof uses neither a geometric flow nor the Bochner identity for harmonic maps.

**3.** **Jian Ge ****葛剑****, Peking University **

**Title:** On the Parallel Axiom

**Abstract:** In this talk, we will discuss a rigidity result for the Riemannian metric satisfying the Parallel Axiom, based on the joint work with L. Guijarro and P. Solorzano.

**4.** **Juanru Gu ****顾娟如****, Zhejiang University of Technology**

**Title****：** Rigidity and sphere theorems of submanifolds

**Abstract****：** In this talk, we will focus on geometry and topology of submanifolds and introduce some results on rigidity and sphere theorems of submanifolds. We will also discuss some new techniques developed in the study of geometry and topology of submanifolds. Finally we will propose some open problems in this area.

**5.** **Xiantao Huang, ****黄显涛****, Sun Yat-Sen University**

**Title:** The asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth

**Abstract:** Suppose (M, g) is a Riemannian manifold with nonnegative Ricci curvature, and let hd(M) be the dimension of the space of harmonic functions with polynomial growth of growth order at most d. Colding and Minicozzi proved that hd(M) is finite. Later on, there are many researches which give better estimates of hd(M). In this talk, we will present the work on asymptotic behavior of hd(M) when d is large. More precisely, suppose that (M, g) has maximal volume growth and its tangent cone at infinity is unique, then when d is sufficiently large, we obtain some estimates of hd(M) in terms of the growth order d, the dimension n and the asymptotic volume ratio of (M, g).

**6. Jing Mao ****毛井****, Hubei University**

**Title:** Some results about spectral analysis and functional inequalities on metric measure spaces

**Abstract:** In this talk, we would like to show some interesting results about spectral estimates and functional inequalities on smooth or non-smooth metric measure spaces, which are based on several works of mine and joint-works with my collaborators.

**7.** **Jiayin Pan ****潘佳垠****, Rutgers University**

**Title:** Milnor conjecture in low dimensions

**Abstract:** Milnor conjectured that any open n-manifold of non-negative Ricci curvature has a finitely generated fundamental group. In this talk, we use equivariant Gromov-Hausdorff convergence and Cheerger-Colding theory to approach Milnor conjecture in low dimensions. We give a proof in dimension 3; in dimension 4, we confirm Milnor conjecture when the universal cover has Euclidean volume growth and unique tangent cone at infinity.

**8.** **Sheng Rao ****饶胜****, Wuhan University**

**Title:** Geometry of logarithmic forms and deformations of complex structures

**Abstract:** We present a new method to solve certain dbar-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a ddbar-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at E1-level, as well as certain injectivity theorem on compact Kahler manifolds.

Furthermore, for a family of logarithmic deformations of complex structures on Kahler manifolds, we construct the extension for any logarithmic (n,q)-form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by differential geometric method.

This talk is based on a recent joint work arXiv: 1708.00097 with Kefeng Liu and Xueyuan Wan.

**9. Fang Wang ****王芳****, Shanghai Jiaotong University**

**Title:** Conformally Compact Einstein Manifolds and Yamabe Invariants

**Abstract:** On conformally compact Einstein manifolds, several types of Yamabe invariants can be defined. I will talk about the relations between these Yamabe invariants in terms of inequalities, and also show some rigidity theorem based on those inequalities.

**10. Guoyi Xu ****徐国义****, Yau Mathematical Sciences Center**

**Title:** When the fundamental group of a Riemannian manifold is finitely generated?

**Abstract:** For every compact Riemannian manifold, it is well known that the fundamental group is finitely generated. For complete non-compact Riemannian manifolds, the fundamental group possibly is not finitely generated. A natural question is: which complete Riemannian manifolds have finitely generated fundamental group? We will survey the progress in this question from Bieberbach, Cheeger-Gromoll, Gromov to more recent work by Kapovitch and Wilking, and my recent work will also be presented. No technical proofs in the talk, some elementary topology and Riemannian geometry knowledge is enough to understand most of the talk.

**11. Chengjie Yu ****余成杰****, Shantou University**

**Title:** Rigidity of isometric embedding into the light cone

**Abstract:** Isometric embedding is a classical subject in differential geometry. It has also important relations with General Relativity on quasilocal mass. In this talk, we will present a recent work joint with Dr. Jian-Liang Liu on the rigidity of isometric embedding into the light cone of the Minkowski spacetime.

**12.** **Huichun Zhang, ****张会春****, Sun Yat-Sen University**

**Title****：** Weyl’s law for eigenvalues on singular contexts

**Abstract:** One of most fundamental theorems in spectral geometry is the Weyl’s law: on any closed ndimensional Riemannian manifold, we have a leading asymptotic for eigenvalues of Laplace operator. In this talk, we will introduce an extension of the Weyl’s law to singular metric spaces with generalized Ricci curvature bounded from below. This is a joint work with Prof. Xi-Ping Zhu.

**13.** **Yongsheng Zhang ****张永胜****, Northeast Normal University**

**Title:** On Lawson-Osserman's construction

**Abstract:** The 1977’ Acta paper by Lawson-Osserman studied the Dirichlet problem for minimal surfaces of high codimensions. Several astonishing results essentially distinct from the case of codimension 1 were obtained there. In particular, they found Lipschitz but non-C1 solutions to the problems associated to Hopf maps between unit spheres. Recently we developed Lawson-Osserman’s constructions and discovered certain interesting new phenomena on the existence, non-uniqueness, non-minimizing and minimizing properties of solutions to related Dirichlet problems. This talk is based on joint works with Xiaowei XU and Ling YANG.