2017 ECNU Winter Workshop

of Geometry and Analysis on Manifolds

Symplectic aspects of polar actions

Chen, Xiaoyang
E-mail: xychen100@tongji.edu.cn
Tongji University

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.

On Eells-Sampson type theorems for subelliptic harmonic maps

Dong, Yuxin
E-mail: yxdong@fudan.edu.cn
Fudan University

Abstract: A sub-Riemannian manifold is a manifold with a subbundle of the tangent bundle and a fiber metric on this subbundle. A Riemannian extension of a sub-Riemannian manifold is a Riemannian metric on the manifold compatible with the fiber metric on the subbundle. One may define an analog of the Dirichlet energy by replacing the L2 norm of the derivative of a map between two manifolds with the L2 norm of the restriction of the derivative to the subbundle when the domain is a sub-Riemannian manifold. A critical map for this energy is called a subelliptic harmonic map. In this talk, by use of a subelliptic heat flow, we establish some Eells-Sampson type existence results for subelliptic harmonic maps when the target Riemannian manifold has non-positive sectional curvature.

Finite time blowup of the n-harmonic flow on n-manifolds

Hong, Min-chun
University of Queensland, Australia

Abstract: In this talk, we generalize the no-neck result of Qing-Tian to show that there is no neck during blowing up for the n-harmonic flow as t goes to infinity. As an application of the no neck result, we settle a conjecture of Hungerbühler by constructing an example to show that the $n$-harmonic map flow on an n-dimensional Riemannian manifold blows up in finite time for n≥ 3. (This is my joint work with Leslie Cheung.)

On the $L_p$ version of Aleksandrov problem

Huang, Yong
Hunan University

Abstract: Based on some basis of convex geometry, and Aleksandrov problem which solved by Aleksandrov at 1942. In this talk, we introduce Lp-Aleksandrov problem, and reduce it to a new Monge-Ampere type equation on sphere S^n. Its weak solution is obtained by using the variational formula of our early work. This is a joint work with Erwin Lutwak,Deane Yang and Gaoyong Zhang.

Structure of noncollapsing Ricci Limit spaces

Jiang, Wenshuai
Hunan University

Abstract: Let us consider $(M^n_i,g_i,p_i)\to (X,d,p)$ in Gromov-Hausdorff sense with $Vol(B_1(p_i))>v>0$ and $Ric>-(n-1)$. We will show that the singular set
$S$ of $X$ is (n-2)-rectifiable. More generally, for $0\leq k<n$, the $k$-stratum $S^k=\{x\in X: \text{no tangent cone at}\ x\ \text{splits a}\ R^{k+1}\ \text{factor}\}$ is $k$-rectifiable. We will also discuss the quantitative estimate of $S^k$. This is joint work with J.Cheeger and A.Naber.

SL(2, C)-Chern-Simons and its gradient flow

Li, Weiping
E-mail: w.li@okstate.edu
Oklahoma State University

Abstract: This is to understand the complexified Chern-Simons theory for the SU(2) instanton Floer theory which fits nice into the topological quantum field theory. We try to use the extra fields to balance the complexified Chern-Simons to make a proper definition of the gradient flow. The construction is formally similar to instanton gauge Floer theory and Donaldson theory in 3- and 4-dimensional manifolds.

Eta Forms and Differential K-Theory

Liu, Bo
E-mail: bliu@math.ecnu.edu.cn
East China Normal University

Abstract: In this talk, I'll study the push-forward map in differential K theory along the proper submersion and the embedding under the model of Bunke-Schick. Furthermore I'll discuss the compatibility of them by extending the Bismut-Zhang embedding formula for the eta invariants to the family case.

Hearing the shape of a trapezoid by its Laplace spectrum

Lu, Zhiqin
E-mail: alexxlu@me.com
U.C. Irvine, USA

Abstract: It is well known that one can “hear” the shape of a triangle drum. For a general polygon, the same problem is surprisingly still open. We shall prove that the shape of a trapezoid is determined by its Neumann eigenvalues. The Dirichlet eigenvalue case will also be discussed. The proof depends on the analysis of singularities of the wave trace, and therefore it is related to the classification of short families of geodesics of a trapezoid. This is the joint work with Hamid Hezari and Julie Rowlett.

On Dirichlet problem for minimal graphs and Lawson-Osserman constructions

Yang, Ling
Fudan University

Abstract: We develop the Lawson-Osserman's works on minimal graphs. Firstly, we construct a constellation of uncountably many Lawson-Osserman spheres, which are minimal in Euclidean spheres and therefore generate Lawson-Osserman cones that correspond to Lipschitz but non-differentiable solutions to the minimal surface system. Then, by the theory of autonomous systems in plane, we find for each Lawson-Osserman cone an entire minimal graph having it as tangent cone at infinity. Further, in addition to the truncated Lawson-Osserman cones, we discover infinitely many analytic solutions to the Dirichlet problem of minimal surfaces system for boundary data induced by certain Lawson-Osserman spheres. As a corollary, those Lawson-Osserman cones are non-minimizing. These behaviors are observed for the first time. This is the joint work with Prof. Xiaowei Xu and Yongsheng Zhang.

Degeneration of Riemannian manifolds with bounded Bakry-Emery Ricci curvature

Zhu, Meng
E-mail: mzhu@math.ecnu.edu.cn
East China Normal University

Abstract: We study the regularity of the Gromov-Hausdorff limits of Riemannian manifolds with bounded Bakry-Emery Ricci curvature, which includes the Ricci soliton and bounded Ricci curvature cases.We generalize Cheeger-Colding-Tian-Naber's results when the manifolds are volume noncollapsing. Our new ingredients here are a Bishop-Gromov type relative volume comparison theorem on the original manifold without involving weight, and proving that the $C^{\alpha}$ harmonic radius can be bounded from below, which has weakened Anderson's result. Our proof of the Codimension 4 Theorem essentially follows the guideline of Cheeger-Naber, but we managed to shorten the proof by using Green's function and a linear algebra argument of R. Bamler. These are joint works with Qi S. Zhang.