2015几何分析学术研讨会

2015年几何分析学术研讨会会议通知

教授：

此致

厦门大学数学科学学院

2015年4月15

1. 会议日程：2015516  会议报到时间

2015年5月17  会议正式开始

2015年5月22  会议结束

2. 会议地点：厦门市白鹭洲大酒店

3. 联系人：郭淑敏，厦门大学数学科学学院

Email

电话：86-592-2580602

韩青  李嘉禹  邱春晖  史宇光  张明智  钟春平

 分工 姓名 联系电话 报到、考察 郭淑敏 13400785041 交通 雷玉娟 18959212796 住宿、用餐 林智雄 13599916563 会场、多媒体 林  煜 18959219403

Schedule

May 17, Sunday,厦门大学数学物理大楼117报告厅

 9:10-9:30 Opening Ceremony 9:40—10:30 徐兴旺 Negative gradient flow and the scalar curvature function 11:00—11:50 葛剑 Distance function to the boundary of manifold

May 17, Sunday,厦门大学数学物理大楼117报告厅

 14:00—14:50 李海中 Self-shrinkers of the mean curvature flow in arbitrary codimension 15:20—16:10 周斌 A potential theory for Weingarten curvature equations 16:20—17:10 宋翀 Skew mean curvature flow

May 18, Monday,厦门大学数学物理大楼117报告厅

 9:00—9:50 戎小春 Manifolds of Ricci curvature and volume of local covering bounded below 10:20—11:10 朱苗苗 Boundary value problems for Dirac equations and the heat flow forDirac-harmonic maps 11:20—11:50 姜旭旻 Boundary expansion for the complex Monge Ampere equation

May 18, Monday,厦门大学数学物理大楼117报告厅

 14:00—14:50 陈群 Omori-Yau maximum principles, V-harmonic maps and self-shrinkers of mean curvature flows 15:20—16:10 张会春 Lipschitz regularity of harmonic maps from Alexandrov spaces to NPC spaces 16:20—17:10 葛化彬 A combinatorial Yamabe problem on two and three dimensional manifolds

May 19, Tuesday,厦门大学数学物理大楼117报告厅

 9:00—9:50 傅吉祥 Teissier's proportionality problem 10:20—11:10 Isoperimetric type problems for quermassintegrale in hyperbolic space 11:20—11:50 楚建春 $C^{2,\alpha}$ estimates for some nonlinear elliptic equations in complex geometry

May 19, Tuesday,厦门大学数学物理大楼117报告厅

 14:00—14:50 盛利 The Exponential Decay of Gluing Maps and Gromov-Witten Invariants 15:20—16:10 刘佳堃 On the uniqueness of $L_p$-Minkowski problems 16:20—17:10 刘磊 Energy quantization and blow up analysis for Dirac-harmnic maps with curvature term

May 20, Wednesday,厦门大学数学物理大楼117报告厅

 9:00—9:50 陈兵龙 A survey on some aspects of the Ricci flow theory 10:20—11:10 华波波 Discrete Harmonic function theory on graphs 11:20—11:50 戴嵩 Lower order tensors in non-Kähler geometry and non-Kähler geometric flow

May 20, Wednesday,厦门大学数学物理大楼117报告厅

 14:00—14:50 张希 Some results on semistable Higgs bundle 15:20—16:10 王枫 Stability in Kähler-Ricci soliton 16:20—17:10 马世光 Constant mean curvature surfaces of Delaunay type along a closed geodesic

May 21, Thursday,厦门大学数学物理大楼117报告厅

 9:00—9:50 关波 Fully nonlinear elliptic equations on Hermitian manifolds 10:20—11:10 陈竟一 Radially symmetric solutions to the graphic Willmore surface equation 11:20—11:50 鲍超 Topology of closed mean convex hypersurfaces with low entropy

Title and Abstract

Topology of closed mean convex hypersurfaces with low entropy

In this presentation, I will show that the closed mean convex hypersurfaces with low entropy are diffeomorphic to round spheres, this work is inpired by the results of Colding, Minicozzi and etc. We will use techniques from mean curvature fow  to get the main result.

A survey on some aspects of the Ricci flow theory

I will report briefly on the development of the Ricci flow during past 30 more years, focusing on my joint works with Xi-Ping Zhu in this field over past 15 years. This includes our joint works on uniqueness theorem, pinching theorem, gap theorems, classification of four-manifolds with positive isotropic curvature, and uniformization theorem in complex differential geometry.

Radially symmetric solutions to the graphic Willmore surface equation

We show that a smooth radially symmetric solution $u$ to the graphic Willmore surface equation is either a constant or the defining function of a half sphere in ${\mathbb R}^3$. In particular, radially symmetric entire Willmore graphs in ${\mathbb R}^3$ must be flat. When $u$ is a smooth radial solution over a punctured disk $D(\rho)\backslash\{0\}$ and is in $C^1(D(\rho))$, we show that there exist a constant $\lambda$ and a function $\beta$ in $C^0(D(\rho))$ such that  $u''(r) =\frac{\lambda}{2}\log r+\beta(r)$; moreover, the graph of $u$ is contained in a graphical region of an inverted catenoid which is uniquely determined by $\lambda$ and $\beta(0)$. It is also shown that a radial solution on the punctured disk extends to a $C^1$ function on the disk when the mean curvature is square integrable.

Omori-Yau maximum principles, V-harmonic maps and self-shrinkers of mean curvature flows

In this talk, we first introduce our recent results on Omori-Yau maximum principles, then, combining with V-harmonic maps, we will give some applications to rigidity problems of self-shrinkers and translating solitons of mean curvature flows.

$C^{2,\alpha}$ estimates for some nonlinear elliptic equations in complex geometry

In this talk, we will consider some nonlinear elliptic equations in complex geometry with Hölder-continuous right hand side. We will present the sharp $C^{2,\alpha}$ estimates for solutions of these equations, assuming a bound on the Laplacian of the solution. Our result is optimal regarding the Hölder exponent.

Lower Order Tensors in non-Kahler Geometry and non-Kahler Geometric Flow

Abstract: In recent years, Streets and Tian introduced a series of curvature flows to study non-Kahler geometry. In this talk, we discuss how to construct second order curvature flows from the viewpoint of tensor. By classifying lower order tensors, we classify second order almost Hermitian curvature flows, under some natural conditions. As a corollary, we show symplectic curvature flow is the unique way to deform almost Kahler structure in some canonical sense.

Teissier's proportionality problem

In this talk, I will talk about Teissier's proportionality problem fortranscendental nef classes over a compact Kählermanifold which says that the equalities in the Khovanskii-Teissier inequalities hold for two nef and big classes if and only if the two classes are proportional. This is a joint work with my student Xiao Jian.

A combinatorial Yamabe problem on two and three dimensional manifolds

We introduce a new combinatorial curvature on two and three dimensional triangulated manifolds, which transforms in the same way as that of the smooth scalar curvature under scaling of the metric and could be used to approximate the Gauss curvature on two dimensional manifolds. Then we use the flow method to study the corresponding constant curvature problem, which is called combinatorial Yamabe problem. This work is joint with XuXu.

Distance function to the boundary of manifold

We focus on the study of the distance function to the boundary of a Riemannian manifold. Various assumptions on curvatures of the Riemannian manifold and mean curvature of the boundary allow us to estimate this function. We will give some applications to eigenvalue estimate and contact metric geometry.

Title: Fully nonlinear elliptic equations on Hermitian manifolds.

We are concerned with a class of fully nonlinear elliptic equations which play important roles of complex geometry and analysis. In the talk we hall focus on second derivative estimates on closed Hermitian manifolds. As an application of our results, we can prove a conjecture of Gauduchun building on results of Tossati and Weinkove.The major part of this talk is based on joint work with Xiaolan Nie.

Discrete Harmonic Function Theory on Graphs

As discrete metric measure spaces, graphs play an important role in both theoretic and applicative way. The combinatorics is particularly helpful to understand the structure of finite graphs. While for infinite graphs, which resemble complete noncompact Riemannian manifolds, these methods are no longer effective, geometric methods survive. We will talk about the discrete harmonic function theory on infinite graphs in this talk.

Boundary expansion for the complex Monge Ampere equation

In this talk, we discuss boundary expansions for complete Kahler metrics in bounded strictly pseudoconvex domains. We discuss the remainder estimates in the context of local finite regularity and the convergence of the expansions under a natural condition that the boundary is analytic.

Self-shrinkers of the mean curvature flow in arbitrary codimension

In this talk, we will report our recent results in the study of the self-shrinkers of the mean curvature  flow in Euclidean space with arbitrary codimension, which include gap theorems about the norm square of the second fundamental form, classification results, rigidity results, F-stability and the volume estimates of self-shinkers.

On the uniqueness of $L_p$-Minkowski problems

In this talk we first give a brief introduction to the $L_p$-Minkowski problem. Then we focus on the uniqueness results and show that in dimension two, either when $p \in [-1,0]$ or when $p \in (0,1)$ in addition to a pinching condition, the solution must be the unit ball. This partially answers a conjecture of Lutwak, Yang and Zhang about the uniqueness of the $L_p$-Minkowski problem in dimension two. Moreover, we give an explicit pinching constant depending only on $p$ when $0<p<1$. This is a recent joint work with Yong Huang and Lu Xu.

Energy quantization and blow up analysis for Dirac-harmnic maps with curvature term

For a sequence of Dirac-harmonic maps with curvature term from a closed Riemannian surface to a general compact Riemannian manifold with uniformly bounded energy, we prove that the energy identities and neckless results hold during the blow up process. This is a joint work with Prof. Jurgen Jost and Prof. Miaomiao Zhu.

Constant mean curvature surfaces of Delaunay type along a closed geodesic

Delaunay surfaces are a kind of periodic surfaces with constant mean curvature in Euclidean space R^3. In this talk I will show how to construct Delaunay type constant mean curvature surfaces along a non degenerate closed geodesic in 3-dim Riemannian manifolds. This is a joint work with Frank Pacard.

Manifolds of Ricci curvature and volume of local covering bounded below

This will be a preliminary report on on-going program with Jiayin Pan in the study of the manifolds in the title.

The Exponential Decay of Gluing Maps and Gromov-Witten Invariants

There had been several different approaches to define Gromov-Witten Invariants for general symplectic manifolds, such as Fukaya-Ono, Li-Tian, Liu-Tian, Ruan, Siebert and etc. The moduli space has various lower strata. We show that the relevant differential form decays in exponential rate near lower strata, then the Gromov-Witten invariants can be defined as an integral over top strata of virtual neighborhood.

Skew Mean Curvature Flow

The skew mean curvature flow or binormal flow, which origins from fluid dynamics,describes the evolution of a codimension two submanifold along the binormal direction. We show the existence of a local solution to the initial value problemof the SMCF of surfaces in Euclidean space R4. As a byproduct, we obtain a uniform Sobolev-type embedding theorem for the second fundamental forms of two dimensional surfaces. I will also talk about the Hasimoto transformation if time permits. This is a joint work with Jun Sun.

Stability in Kähler-Ricci soliton

The notion of K-stable was introduced by Tian to study the existence of Kahler-Einstein metric. This was generalized by Donaldson in a pure algebraic way. For Kähler-Ricci soliton,  Berman gave a definition of modified K-stable. From the equivariant point of view, we will see the similarity between these two notions of stability.

Isoperimetric type problems for quermassintegrale in hyperbolic space

In this talk, I will start from the classical Alexandrov-Fenchel type inequalities for convex domains in Euclidean space and turn to the recent development of such kind of inequalities in hyperbolic space. We show that, for the class of horospherical convex hypersurfaces in hyperbolic space whose $l$-thquermassintegral is a constant, the $k$-th $(k>l)$ quermassintegrale attains its minimum at geodesic spheres. As a byproduct several isoperimetric type problems about curvature integrals are solved in the class of horo-convex hypersurfaces. Our proof relies on the result of quermassintegral preserving curvature flow. This is joint work with Guofang Wang.

Negative Gradient Flow and the scalar curvature function

In this talk, I will update the information on the scalar curvature function problem in a conformal class by the negative gradient flow method.

Lipschitz Regularity of Harmonic Maps from Alexandrov Spaces to NPC Spaces

In 1997, J. Jost and F. H. Lin proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to a non-positivecurvature (NPC) metric space is locally Holder continuous. In [37], F. H. Linconjectured that the Holder continuity can be improved to Lipschitz continuity. J. Jost also asked a similar problem. In this talk, we will introduce a resolution to this Lin’s problem. This is a joint work with Prof. Xi-Ping Zhu.

Some results on semistable Higgs bundle

In this talk, I will talk about the heat flow on Higgs bundle and introduce our recent works (joint with Jiayu Li and Yechi Nie) on semistable Higgs bundle.

A potential theory for Weingarten curvature equations

In this talk, we will introduce a potential theory for Weingarten curvature (or k-curvature) equation, which can also be seen as a PDE approach to curvature measures. In the case of k=1, we extend the mean curvature measure to signed measures. The related prescribed mean curvature equation will be solved. When k>1, we assign a measure to a bounded, upper semicontinuous function which is strictly subharmonic with respect to the k-curvature operator, and establish the weak continuity of the measure.

Boundary Value Problems for Dirac Equations and the Heat Flow for

Dirac-harmonic Maps

In this talk, we shall first discuss the existence, uniqueness andimproved elliptic estimates for Dirac equations under a class of localelliptic boundary conditions. Then we introduce a heat flow approach tothe existence of Dirac-harmonic maps from Riemannian spin manifolds withboundary and show the short time existence of this flow. This is a joint work with Qun Chen, Juergen Jost and Linlin Sun.

1、机场T4候机楼→白鹭洲大酒店：

①乘坐的士：约40元；

②乘坐公交：乘坐乘BRT1线至“二市站”，换乘L5路公交车至“天湖苑站”。

2、机场T3候机楼→白鹭洲大酒店：

①乘坐的士：约30元；

②乘坐公交：乘L19路至“县后站”下车，换乘BRT1线至“二市站”，换乘L5路公交车至“天湖苑站”。

3、火车站（厦门岛内）→白鹭洲大酒店：

①、乘坐的士：约12元；

②、乘坐公交：乘坐122路公交车至“非矿站”下车；或乘坐958路至“天湖苑站”。

4、厦门北站（岛外）→白鹭洲大酒店：

①、乘坐的士：约60元；

②、乘坐公交：乘BRT1线至“二市站”下车， 换乘坐L5路公交车至“天湖苑站”下车。

5、机场T4候机楼→厦门大学海韵园（会场）：

①、乘坐的士：约50元；

②、乘坐公交车：乘BRT1线至“火车站”下车，换乘旅游2线至“海韵”。

6、机场T3候机楼→厦门大学海韵园（会场）

①、乘坐的士：约50元；

②、乘坐公交车：乘L19路至“县后站”下车，换乘BRT1线至“火车站”，换乘旅游2线至“海韵站”下车。

7、火车站（厦门岛内）→厦门大学海韵园（会场）：

①、乘坐的士：约20元；

②、乘坐公交车：乘坐旅游2线至“海韵站”下车。

8、厦门北站（岛外）→厦门大学海韵园（会场）

①、乘坐的士：约80元；

②、乘坐公交：乘BRT1线至“火车站”，换乘旅游2线至“海韵”。

9、厦门大学海韵园（会场）→白鹭洲大酒店：

①、乘坐的士：约20元；

②、乘坐公交：“珍珠湾站”乘坐87路至“天湖苑站”。