2016 ECNU Winter Workshop

of Geometry and Analysis on Manifolds

几何分析中的凸体理论

Leng, Gangsong

E-mail:lenggangsong@163.com

Shanghai University, P.R.China

TBA

Lu, Zhiqin

E-mail:fht@nankai.edu.cn

U.C. Irvine, USA

非线性椭圆方程的Neuman问题以及对应抛物平移解

Ma, Xinan

E-mail:xinan@ustc.edu.cn

University of Science and Technology of China, P.R.China

我们研究平均曲率方程与k-Hessian方程的Neumann边值问题，我们引进梯度估计与二阶导数估计的新方法，得到了存在性定理。在区域是凸的前提下我们也研究他们的抛物对应，如对于非参数平均曲率流我们得到一般Neumann边值问题的平移解。这是与徐金菊，邱国寰，王培合，韦韡等的合作工作。

On the growth of von Neumann dimension of harmonic spaces of semipositive line bundles over covering manifolds

Wang, Huan

E-mail:huanwang2016@hotmail.com

Universität zu Köln, German

We study the harmonic space of line bundle valued forms over a covering manifold with a discrete group action, and obtain an asymptotic estimate for the von Neumann dimension of the space of harmonic (n,q)-forms with values in high tensor powers of a semipositive line bundle. In particular, we estimate the von Neumann dimension of the corresponding reduced L2-Dolbeault cohomology group. The main tool is a local estimate of the pointwise norm of harmonic forms with values in semipositive line bundles over Hermitian manifolds.

From symplectic reduction to equivariant spectral geometry

Wang, Zuoqin

E-mail:wangzuoq@ustc.edu.cn

University of Science and Technology of China, P.R.China

Let $G$ be a compact Lie group acting as isometries on a compact Riemannian manifold $(M, g)$. Then each eigenspace of the Laplace-Beltrami operator is a representation of $G$, from which one gets a much finer structure of the Laplacian spectrum. In this talk I will explain the role of symplectic reduction in this equivariant spectral theory, and how to use the equivariant spectrum to recover Schrodinger potentials on symplectic toric manifolds. This is a joint work with V. Guillemin.

Heat kernel estimate along Ricci-harmonic flow

Wu, Guoqiang

E-mail: gqwu@math.ecnu.edu.cn

East China Normal University, P.R.China

In this talk we discuss Ricci-harmonic flow on which the scalar curvature is bounded. At first, we establish a time derivative bound for solution to the heat equation, based on this, we derive the distance distortion estimate and the existence of a cutoff function. At last we use these to get the heat kernel upper bound and lower bound along Ricci-harmonic flow. This is joint with Prof Yi Li from Shanghai Jiaotong University.

A uniﬁed treatment for Lp Brunn-Minkowski type inequalities

Xiong, Ge

E-mail: xiongge@tongji.edu.cn

Tongji University, P.R.China

A unified approach used to generalize classical Brunn-Minkowski type inequalities to Lp Brunn-Minkowski type inequalities, called the Lp transference principle, is refined in this paper. As illustrations of the effectiveness and practicability of this method, several new Lp Brunn-Minkowski type inequalities concerning the mixed volume, moment of inertia, quermassintegral, projection body and capacity are established.

Regularity and blow-up analysis at the free boundary for approximate harmonic maps from surfaces

Zhu, Miaomiao

E-mail:mizhu@sjtu.edu.cn

Shanghai Jiaotong University, P.R.China

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 harmonic map flow with free boundary. Also, we shall briefly discuss some related works. These are joint works with Jürgen Jost and Lei Liu.