### 2018年厦门几何分析学术研讨会

 日期 时间 事项 1月13日 9:00-9:20 开幕式，合影（主持人：） 主持人：徐兴旺 9:20-10:05 朱小华（北京大学）: $G$-Sasaki manifolds and $K$-energy 10:10-10:55 葛化彬（北京交通大学）：Characterizing   ideal polyhedra in hyperbolic 3-space by combinatorial and angle structure 10:55-11:15 休息 11:15-12:00 李宇翔（清华大学）: John-Nirenberg Inequality and  Collapse    in Conformal Geometry 12:00-14:00 午餐、休息 主持人：夏  超 14:00-14:45 张  希（中国科学技术大学）: Some   results on complex Monge-Amp\ere equation 14:50-15:35 江文帅（浙江大学）：The Structure of   Noncollapsing Ricci Limit Spaces 15:35-15:55 休息 15:55-16:40 陈学长（南京大学）: Boundary Yamabe problem and rigidity   theorems of Poincare-Einstein manifolds 16:45-17:30 杨  波（厦门大学）: On the space of Hermitian connections   and else 1月14日 主持人：邱春晖 9:00-9:45 陈兵龙（中山大学）: On Kahler hyperbolicity  or    non-ellipticity and fundamental groups 9:50-10:35 周恒宇（中山大学）: Hypersurfaces   with prescribed mean curvature in product manifolds. 10:35-10:55 休息 10:55-11:40 陈  群（武汉大学）: Existence   for Dirac equations and Dirac-harmonic maps 11:40-14:00 午餐、休息 主持人：宋  翀 14:00-14:45 张立群（中国科学院）: On the regularity of weak solution   of parabolic equations 14:50-15:35 戴  嵩(天津大学): Dominations   in cyclic Higgs bundles 15:35-15:55 休息 15:55-16:40 隋哲楠（哈尔滨工业大学）: Complete   Conformally Flat Metrics of Prescribed k-Ricci Curvature on Exterior Domains 16:45-17:30 贺  飞（厦门大学）: Short-time   existence of Ricci flow on noncompact manifolds 1月15日 主持人：关  波 9:00-9:45 莫小欢（北京大学）：The Geometry of   Spherically Symmetric Finsler Manifolds 9:50-10:35 王  鹏（同济大学）: Morse Index and Willmore Stability   of Minimal Surfaces in Spheres 10:35-10:55 茶歇 10:55-11:40 麻希南（中国科学技术大学）: Mean   curvature flow with Neumann boundary problem

On Kahler hyperbolicity or non-ellipticity and fundamental groups

Abstract: We prove that a compact Kahler manifold with torsion pi_2 is Kahler hyperbolic or Kahler non-elliptic if the fundamental group pi_1 is hyperbolic or satisfies a quadratic radial isoperimetric inequality. The latter case includes many interesting groups, e.g. CAT(0) groups, automatic groups.  In particular, the signed Euler characteristic of such manifold is strictly positive or nonnegative respectively. The paper is a joint work with Xiaokui Yang.

Existence for Dirac equations and Dirac-harmonic maps

Abstract:  Dirac equation on manifolds is a geometric analytic model which has very important influence to both mathematics and physics, Atiyah-Singer index theorems for Dirac operators constitutes a cornerstone of modern mathematics. While index theorems and Fredholm theorems provide criteria for existence of solutions, for given elliptic boundary problems, one often needs more precise results for the existence of solutions. Dirac-harmonic map is a coupled system of a Dirac equation and harmonic map type equation, the existence for which has been a very difficult problem for years. In this talk, we will first discuss existence and uniqueness for solutions of Dirac equations satisfying a class of local elliptic boundary value conditions. Then we will discuss existence for Dirac-harmonic maps and their heat flows.

Boundary Yamabe problem and rigidity theorems of Poincare-Einstein manifolds

Abstract: We first prove the rigidity result in the equality case of Gursky-Han's inequality on Poincare-Einstein inequality, which involves boundary Yamabe problem of constant scalar curvature conformal metric with minimal boundary. Next we prove anther conformally invariant inequality on Poincare-Einstein manifolds, as well as the rigidity result. This relates to a compactification by using scalar flat conformal metric with constant boundary mean curvature. This is joint with Mijia Lai and Fang Wang.

Dominations in cyclic Higgs bundles

Abstract: In this talk, we will introduce the higher Teichmuller theory from the viewpoint of the Higgs bundles. Given a G-Higgs bundle over a Riemann surface \Sigma, there is a corresponding harmonic map f from \tilde{\Sigma} to the symmetric space G/K. We study a subfamily in the space of the Higgs bundles, called the cyclic Higgs bundles. By introducing a maximum principle for the elliptic systems, we obtain some domination results about the pullback metric of f and the curvature.  It is a joint work with Qiongling Li, https://arxiv.org/abs/1710.10725 .

Characterizing ideal polyhedra in hyperbolic 3-space by combinatorial and angle structure

AbstractAround 1980, Thurston showed that “almost every” 3-manifold admits a complete hyperbolic metric.

To get such a metric, he proposed to ideally triangulate the manifold and realize each tetrahedron as a hyperbolic ideal tetrahedron. He also gave a system of gluing equations in the shape parameter of these ideal tetrahedrons, whose solution corresponds to the complete hyperbolic metric.

In the 1990s, Casson discovered a powerful technique for solving Thurston's gluing equations. The main idea is to study the combinatorial structure of the triangulation and the dihedral angle structure of each tetrahedron.

Following Casson's program, Rivin completely describes all convex ideal polyhedra by combinatorial and angle structures.

In this talk, we shall use combinatorial Ricci flow methods, initiated by Bennett Chow and Luo Feng, to approach Casson-Rivin's program. We shall extend Koebe-Andreev-Thurston's Circle Pattern Theorem, Rivin's theorem on ideal hyperbolic polyhedra and Chow-Luo's theory on combinatorial Ricci flows. Our results suggest an algorithm exponentially fast to find (ideal) circle patterns and ideal hyperbolic tetrahedrons with the given combinatorial type and dihedral angles. This is joint work with Hua Bobo and Zhou Ze.

Short-time existence of Ricci flow on noncompact manifolds

Abstract: The Ricci flow is a canonical geometric evolution equation that has had a profound influence on modern geometric analysis. On a closed manifold the Ricci flow always exists for any smooth initial metric. However, when the initial manifold is noncompact with unbounded curvature, even the short-time existence of Ricci flow is not well-understood yet. Recently there have been some interesting progress on this problem by many authors. This talk will discuss recent existence results and their applications.

The Structure of Noncollapsing Ricci Limit Spaces

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

John-Nirenberg Inequality and  Collapse  in Conformal Geometry

Abstract: Let $g$ be a metric over $B$, and $g_k=u_k^\frac{4}{n-2}g$. We assume $\|R(g_k)\|_{L^p}<C$, where $R$ is the scalar curvature and $p>\frac{n}{2}$. We will use John-Nirenberg inequality to prove that if $vol(B,g_k)\rightarrow 0$, then there exists $c_k\rightarrow +\infty$, such that $c_ku_k$ converges to a positive function weakly in $W^{2,p}_{loc}(B)$. As an application, we will study the bubble tree convergence of a conformal metric sequence with integral-bounded scalar curvature.

Mean curvature flow with Neumann boundary problem

Abstract: We study nonparametric surfaces over strictly convex bounded domain in $\mathbb{R}^n$, which are evolving by the mean curvature flow with Neumann boundary value. We prove that solutions converge to the ones moving only by translation.

And we will prove the existence and uniqueness of the constant mean curvature equation with Neumann boundary value on strictly convex bounded domain. This is a joint work with P.Wang and W. Wei.

The Geometry of Spherically Symmetric Finsler Manifolds

Abstract: In this lecture we discuss geometry of spherically symmetric Finsler manifolds. We describe non-trivial examples of spherically symmetric Finsler metrics satisfying different curvature conditions. We review recent results in differential geometry with spherically symmetric Finsler metrics.

In particular, we find a sufficient and necessary condition for a spherically symmetric Finsler metric to be a GDW-metric.

As its application, we show that a certain class of spherically symmetric Finsler metrics with are Douglas metrics if they are GDW-metrics, generalized a theorem previously only known in the case of Weyl metrics.

Complete Conformally Flat Metrics of Prescribed k-Ricci Curvature on Exterior Domains

Abstract: The existence of complete conformally flat metrics with negative scalar curvature on exterior domains was systematically studied by Loewner, Nirenberg and Ni, which is the simplest case of the so called noncompact Yamabe problem. In this talk, we will address its fully nonlinear generalization. We show the existence of a complete conformally flat metric with constant k-Ricci curvature when the boundary of the domain consists of smooth compact hypersurfaces. Under the assumption of a global subsolution, the boundary can be more general. We also discover a family of complete conformally flat metrics when the prescribing function is smooth positive satisfying certain decay condition on Euclidean space. The talk is based on joint work with Bo Guan.

Morse Index and Willmore Stability of Minimal Surfaces in Spheres

Abstract: We aim at the Willlmore conjecture in higher co-dimension. It is natural to ask whether the Clifford torus is Willmore stable when the co-dimension increases and whether there are other Willmore stable tori or not.  We answer these problems for minimal surfaces in $S^n$, by showing that the Clifford torus in $S^3$ and the equilateral Itoh--Montiel--Ros torus in $S^5$ are the only Willmore stable minimal tori in arbitrary higher co-dimension. Moreover, the Clifford torus is the only minimal torus (locally) minimizing the Willmore energy in arbitrary higher codimension. And the equilateral Itoh--Montiel--Ros torus is a constrained-Willmore (local) minimizer, but not a Willmore (local) minimizer.  We also generalize Urbano's Theorem to minimal tori in $S^4$ by showing that a minimal torus in $S^4$ has index at least $6$ and the equality holds if and only if it is the Clifford torus. This is a joint work with Prof. Rob Kusner (UMass Amherst).

On the space of Hermitian connections and else

Abstract: The space of Hermitian connections on a given complex manifold is infinite-dimensional in general. In this talk, we present some results on flat Hermitian manifolds.

On the regularity of weak solution of parabolic equations

Abstract: We review the proof of regularty of weak solutions for heat equations and then consider a class of ultraparabolic differential equations with measurable coefficients and lower order terms.  It is proved that weak solutions to the equation are Holder continuous,which generalizes the classic results of parabolic equations with general coefficients. The main observation is a weak Sobolev inequality and Poincare inequality related to weak sub-solutions. This is a jointed work with Wang Wendong.

Some results on complex Monge-Amp\ere equation

AbstractThe complex Monge-Amp\ere equation has been the subject of intensive studies in the past forty years, since its significant applications in complex analysis and complex geometry. In this talk, I will introduce our recent work on regularity estimates of the complex Monge-Amp\ere equation and its applications in K\"ahler geometry and Sasakian geometry. These work are joint with S.Dinew, P.F.Guan, C.Li, J.Y.Li and X.W.Zhang.

$G$-Sasaki manifolds and $K$-energy

Abstract: In this talk, I will introduce a class of Sasaki manifold with a reductive $G$-group action, called $G$-Sasaki manifolds.  By reducing $K$-energy to a functional defined on a class of convex functions on a moment polytope, we can give a criterion for the properness of $K$-energy. In particular, we give a sufficient and nessary condition for the existence of Sasaki Einstein metrics on a $G$-Sasaki manifold. As an application, we found some new Sasaki Einstein manifolds. This is a joint work with Dr. Yan Li.

Hypersurfaces with prescribed mean curvature in product manifolds

Abstract: In this talk we study the existence of $n$-dimensional hypersurface with $C^1$ mean curvature function in warped product manifolds under barrier conditions. With those barrier conditions we construct a family of graphs satisfying auxiliary mean curvature equations. These graphs will converge to an almost minimal boundary in the sense of varifold.  Applying the regularity of the almost minimal boundary, this limit is smooth, embedded satisfying prescribed mean curvature when $n\leq 6$. Similar schemes also work in warped product manifolds and cylinders.