2017 ECNU Summer Workshop

of Geometry and Analysis on Manifolds

Quantitative maximal volume entropy rigidity with Ricci curvature bounded below

Chen, Lina
Capital Normal University, P.R.China
Abstract: For a compact manifold, the volume entropy always exists. Ledrappier-Wang in 2011 showed that with Ricci curvature bounded below by –(n-1), if a manifold achieves the maximal volume entropy n-1, then it is isometric to a hyperbolic manifold. In this talk, I will report a work where we showed that if the volume entropy is almost maximal, then the manifold is diffeomorphic and Gromov-Hausdorff close to a hyperbolic manifold. This is a joint work with Professor Xiaochun Rong and Shicheng Xu.

Convergence of Yamabe flow with ball packings

Ge, Huabin
E-mail: hbge@bjtu.edu.cn
Beijing Jiaotong University, P.R.China
Abstract:Different with the smooth case, the discrete Yamabe flow will generally develop singularities in finite time. However, we can extend the flow through singularities. We also prove that any ball packing can be deformed exponentially fast either to one with constant curvature or to a point along the extended flow. Moreover, in some situations, by proving a Harnack estimate, we show the extend flow indeed converges to a constant curvature packing. This is joint work with Wenshuai Jiang.

Curvature estimates for self-shrinkers and their applications

Guang, Qiang
University of California, Santa Barbara
Abstract:Self-shrinkers model the singularities of the mean curvature flow. In this talk, I will first discuss curvature estimates for almost stable self-shrinkers and mean-convex shrinkers. As applications, the curvature estimates imply a quantitative version of Bernstein-type result for shrinkers and a rigidity result of mean-convex shrinkers. Then I will discuss a curvature estimate for shrinkers with small entropy. A direct corollary of this estimate is a compactness result for shrinkers with small entropy.

Combinatorial curvature for planar graphs

Hua, Bobo
Fudan University and Shanghai Center
Abstract:The combinatorial curvature of a planar graph is defined as the generalized Gaussian curvature of its polygonal surface with a piecewise flat metric. We will show that the total curvature of a planar graph, whose faces are isometric to regular polygons in the Euclidean plane, with nonnegative combinatorial curvature is an integral multiple of 1/6*\pi. This is a joint work with Yanhui Su.

Fundamental gap of convex domains in the sphere

He, Chenxu
University of California, Riverside, USA
Abstract: For a bounded convex domain on a Riemannian manifold, the fundamental gap is the difference of the first two non-trivial Dirichlet eigenvalues. In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap conjecture for convex domains in the Euclidean space, showing that the gap is at least as large as the one for a one-dimensional model. They also conjectured that similar results hold for spaces with constant sectional curvature. Very recently, on the unit sphere, Seto-Wang-Wei proved that the fundamental gap is greater than the gap of the one dimensional sphere model, in particular, ≥ 3 π^2/D^2 (n ≥ 3), provided the diameter of the domain D ≤ π/2. In a joint work with Guofang Wei at UCSB, we extend Seto-Wang-Wei’s lower bound estimate to all convex domains in the hemisphere.

Monge-Ampere equations on the sphere

Wang, Xujia
Australia National University
Abstract: There are a number of geometric and physical problems which can be reduced to the study of the Monge-Ampere equation on the sphere, including the Aleksandrov problem, the Minkowski problem, the p-Minkowski problem, and the dual Minkowski problem. In this talk we give a brief review of these problems and discuss recent development on these problems.

Metrics with non-negative Ricci curvature on convex three-manifolds

Wu, Haotian
University of Sydney, Australia
Abstract:We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path-connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, we show the existence of properly embedded free boundary minimal annulus on any three-ball with non-negative Ricci curvature and strictly convex boundary. This is joint work with Antonio Ache and Davi Maximo.

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

Xu, Guoyi
Tsinghua University, P.R.China
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.

Isoparametric theory and its applications

Yan, Wenjiao
Beijing Normal University, P.R.China
Abstract: In this talk, I will give a brief introduction of isoparametric foliation and some of its applications. The talk is based on joint work with Jianquan Ge and Zizhou Tang.

Positive scalar curvature and the Euler class

Yu, Jianqing
University of Science and Technology of China
Abstract: In this talk, we present the following generalization of the classical Lichnerowicz vanishing theorem: if $F$ is an oriented real flat vector bundle over a closed spin manifold $M$ such that $TM$ carries a metric of positive scalar curvature, then $\langle \widehat A(TM)\,e(F),[M] \rangle=0$, where $\widehat A(TM)$ denotes the Hirzebruch $\widehat A$-class of $TM$ and $e(F)$ is the Euler class of $F$. This is joint work with Weiping Zhang.

Positive scalar curvature and enlargeability

Zhang, Weiping
Chern Institute of Mathematics, P.R.China
Abstract: A famous theorem of Gromov-Lawson states that for any closed spin manifold M of dimension n, the connected sum M#T^n does not carry a metric of positive scalar curvature. We present a potential generalization of this result to the case where M is nonspin. This is a joint work with Guangxiang Su.

Analytic Minimal Model Program through continuity method

Zhang, Zhenlei
Capital Normal University
Abstract: In this talk I will discuss the continuity method approach to the Analytic Minimal Model Program, introduced by La Nave and Tian. The results are based on the joint work with La Nave, Tian and Y. Zhang.

Volume comparison theorem on smooth metric measure spaces

Zhou, Detang
Universidade Federal Fulminense
Abstract: The volume estimates play an important role in Riemannian geometry. Since the works of Wei and Wylie on smooth metric measure spaces, some new volume comparison theorems have been obtained in the recent years. In this talk I will discuss several different proofs of generalized volume comparison theorems.

Minkowski problem and Minkowski isoperimetric inequalities

Zhou, Jiazu
Southwest University, P.R.China
Abstract: The isoperimetric problem is to determine a plane figure of the largest possible area with boundary of a given length and was known in the Ancient Greece. The first mathematically rigorous proof of the isoperimetric problem was obtained only in the 19th century by Weierstrass based on early works of Bernoulli, Euler and Lagrange. The isoperimetric problem is characterized by the isoperimetric inequality. The Minkowski problem is another known problem that is closely related to the isoperimetric inequality, Minkowski inequality, and Brunn-Minkowski inequality in integral geometry and convex geometry.
Originally, the Minkowski problem asks for the construction of a strictly convex compact surface $S$ in the Euclidean space $R^3$ whose Gaussian curvature is specified. More precisely, for a strictly positive real function $f$ defined on the unit sphere $S^2$ in $R^3$, what is necessary and sufficient condition(s) so that there is a convex surface $S$ with Gaussian curvature $f(n(x))$ at the point $x\in S$, where $n(x)$ denotes the normal to $S$ at $x$.
We will introduce some recent exciting results and seek related applications in global geometry, analysis, PDE and other mathematical branches if time allowed.

Scattering matrix and analytic torsion

Zhu, Jialin
Fudan University and Shanghai Center
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 L2-metric on de Rham cohomology. As an application, we give a pure analytic proof of the gluing formula for analytic torsion. This is a joint work with Martin Puchol and Yeping Zhang.