# 2016 ECNU Winter Workshop

## 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

## 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.