2019 ECNU Summer Workshop

of Geometry and Analysis on Manifolds


Guan, Bo
Ohio State University, U.S.A.
Abstract: TBA

Gromov-Hausdorff limits of Kahler manifolds with Ricci curvature lower bound

Liu, Gang
E-mail: gang.liu@northwestern.edu
Northwest University, U.S.A.
Abstract:A fundamental result of Donaldson-Sun states that non-collapsed Gromov-Hausdorff limits of polarized Kahler manifolds, with 2-sided Ricci curvature bounds, are normal projective varieties. We extend their approach to the setting where only a lower bound for the Ricci curvature is assumed. More precisely, we show that non-collapsed Gromov-Hausdorff limits of polarized Kahler manifolds, with Ricci curvature bounded below, are normal projective varieties. In addition the metric singularities are precisely given by a countable union of analytic subvarieties. This is a joint work with Gabor Szekelyhidi.

Domination results for harmonic maps in higher Teichmüller theory

Li, Qiongling
Chern Institute of Mathematics, P.R.China
Abstract:In this talk, we study the harmonic maps in higher Teichmüller theory from the viewpoint of the Higgs bundles. Let X=(S,J) be a closed Riemann surface with genus at least 2. The non-abelian Hodge theory gives a correspondence between the moduli space of representations of the fundamental group of a surface S into a Lie group G with the moduli space of G-Higgs bundles over the Riemann surface X. The correspondence is through looking for an equivariant harmonic map from X to the symmetric space associated to G. Hitchin representations are an important class of representations of fundamental groups of closed hyperbolic surfaces into PSL(n,R), at the heart of higher Teichmüller theory. We discover some geometric properties of such harmonic maps for Hitchin representations or more general representations by using Higgs bundles techniques.

Semi-local simple connectedness of non-collapsing Ricci limit spaces

Pan, Jiayin
University of California, Santa Barbara, U.S.A.
Abstract:We prove that any non-collapsing Ricci limit space is semi-locally simply connected. This is joint work with Guofang Wei.

Heegaard splittings on 3-manifolds: a survey

Qiu, Ruifeng
East China Normal University, P.R.China
Abstract: Let M be a closed, orientable 3-manifold, then there exists a closed surface which cuts M into two handlebodies. This structure on 3-manifold is called Heegaard splitting. In this talk, I will introduce some classical results on Heegaard splitting and its applications.

Green's function estimates and applications

Sung, Chiung-Jue Anna
National Tsing Hua University
Abstract: In this talk, we intend to explain some estimates for the Green's function on complete manifolds admitting a weighted Poincare inequality. Applications will also be mentioned. This is a joint work with Ovidiu Munteanu and Jiaping Wang.

Topology of gradient Ricci solitons

Wang, Jiaping
University of Minnesota, Twins Cities, U.S.A.
Abstract:The talk mainly concerns the issue of connectedness at infinity for gradient Ricci solitons. Ricci solitons are precisely the self-similar solutions to the Ricci flows. They play an important role in the singularity analysis of Ricci flows and are of interest of themselves. This is joint work with Ovidiu Munteanu.

Escobar’s conjecture on lower bound for first Steklov eigenvalue

Xia, Chao
E-mail: chaoxia@xmu.edu.cn
Xiamen University, P.R.China
Abstract: It was conjectured by Escobar in 1999 that for a smooth compact Riemannian manifold with boundary, which has nonnegative Ricci curvature and boundary principal curvatures bounded below by some c>0, the first Steklov eigenvalue is greater than or equal to c with equality holding only on isometrically Euclidean balls with radius 1/c. In this talk, we present a resolution to this conjecture in the case of nonnegative sectional curvature. This is a joint work with Changwei Xiong at ANU.

Solutions to the equations from the conformal geometry

Xu, Lu
Hunan University, P.R.China
Abstract: We solve the Gursky-Streets equations with uniform $C^{1, 1}$ estimates for $2k\leq n$. An important new ingredient is to show the concavity of the operator which holds for all $k\leq n$. Our proof of the concavity heavily relies on Garding's theory of hyperbolic polynomials and results from the theory of real roots for (interlacing) polynomials. Together with this concavity, we are able to solve the equation with the uniform $C^{1, 1}$ \emph{a priori estimates} for all the cases $n\geq 2k$. Moreover, we establish the uniqueness of the solution to the degenerate equations for the first time. TBA

A few properties of global solutions of the heat equation on Euclidean space and some manifolds

Zhang, Qi
University of California,Riverside, U.S.A.
Abstract:We report some recent results on Martin type representation formulas for ancient solutions of the heat equation and dimension estimates of the space of these solutions under some growth assumptions. We will also present a new observation on the time analyticity of solutions of the heat equation under natural growth conditions. One application is a solvability condition of the backward heat equation, i.e. under what condition can one turn back the clock in a diffusion process. Part of the results are joint work with Fanghua Lin and Hongjie Dong.

Unstability of Kaehler-Ricci flow

Zhu, Xiaohua
Peking University, P.R.China
Abstract: In this talk, we will show that there exists a Fano manifold with admitting a Kaehler-Ricci soliton on which the Kaehler-Ricci flow is unstable for Kaehler metrics (the complex structure may vary) in the first Chern class. As a consequence, the second variation of Perelman's entropy on this manifold is not stable for Kaehler metrics in the first Chern class . The situation is totally different on Kaehler-Einstein manifolds on which the second variation of Perelman's entropy is always stable.