the G_A seminar

## The Geometric Analysis Seminar at School of Mathematical Sciences, Xiamen University

The geometric analysis seminar covers various topics on geometry.

## 2017-2018 2nd Semester (2018 Spring)

Schedule Rooms Topics
Tue 02/27 2:30 PM 数理楼661 Organizational meeting.
Fri 03/16 2:00 PM 实验楼105 Chong Song (Xiamen University) "An Energy Method for Uniqueness of Geometric flows"

Abstract: In this talk, I will introduce an energy method for solving uniqueness problems of geometric flows on manifolds. The basic idea is to derive a Gronwall-type inequality for certain geometric energy functionals which describe the intrinsic distance of two solutions. In particular, we use parallel transportations to compare solutions and improve the estimates. I will use the Schrodinger flow as an example and introduce its application to various type of geometric flows.
Fri 03/16 3:30 PM 实验楼105 Yen-Chang Huang (Xinyang Normal University) "A problem of existence of horizontal envelops in the 3D-Heisenberg group and its applications"

Abstract: One of interesting problems in classical geometry is to find the envelope for a family of lines or hypersurfaces in the Euclidean spaces and several applications to Economics and Mathematical Optimization have been developed. After a review of our previous works for finding the pseudohermitian invariants in CR geometry, we will show the necessary and sufficient conditions for the existence of horizontal envelops in the 3D-Heisenberg group by using the standard techniques in Integral Geometry. We obtain a method to construct horizontal envelopes from the given ones and characterize the solutions satisfying the construction. The similar results can be generalized to the higher dimensional Heisenberg groups.
Tue 03/20 2:30 PM 数理楼661 Jinhua Wang (Xiamen University) "An overview: Einstein spaces as attractors for the Einstein flow"

Abstract: I will talk about some stability results in GR, mainly referring to the work by Andersson and Moncrief in Journal of Differential Geometry 89 (2011).
In this paper the authors prove a global existence theorem, in the direction of cosmological expansion, for sufficiently small perturbations of a family of n + 1-dimensional, spatially compact spacetimes, which generalizes the k = −1 FLRW vacuum spacetime. The background spacetimes considered are Lorentz cones over negative Einstein spaces of dimension $n \geq 3$.
These results also demonstrate causal geodesic completeness of the perturbed spacetimes, in the expanding direction, and show that the scale-free geometry converges toward an element in the moduli space of Einstein geometries, with a rate of decay depending on the stability properties of the Einstein geometry.
Fri 03/23 2:00PM (NOT the usual time!) 实验楼105 Xiang Ma (Peking University) "Geometry of pseudo-convex submanifolds in a pseudo-Euclidean space"

Abstract: 伪欧氏空间$\mathbb{R}^{m+1,p}$ 中的$m$维伪凸子流形，是法丛值的第二基本形式 $II(v,v)$ 恒取类空向量的类空子流形，它的法空间中带有一个Lorentz度量（1正p负）。它可以看作是欧氏空间中的凸超曲面（卵形面）的自然推广。直观上，它象是$m+1$维欧氏空间中的凸超曲面的一个微扰。我们先介绍一系列简单的引理，既说明它与凸曲面的类似，也为获得更深入的结果准备技术工具。我们证明它们的截面曲率恒正，并获得了关于全曲率的若干Fenchel 型（反向）不等式。我们还将报导以下结果：在非常弱且自然的条件下，以一个闭的伪凸子流形$\gamma^m$为边界的Plateau问题有解，也就是说，存在一个极大类空$m+1$维子流形$\M^{m+1}$以前者为边界。时间允许的话，我们将简单提及有关概念和结果到离散情形（多边形和多面形）的推广。部分结果是与叶楠博士、张栋博士合作的成果。
Note: At 2:00 PM there is a warm-up talk by the speaker on basics on Lorentz spaces, the research talk begins shortly after the warm-up talk.
Fri 03/23 4:00 PM (NOT the usual time!) 实验楼105 Peng Wang (Tongji University) "On the Morse index of minimal tori in S^4"

Abstract: Urbano's Theorem plays an important geometric role in the proof of Willmore conjecture, which states that a non-totally-geodesic closed minimal surface x in S^3 has index at least 5 and it is congruent to the Clifford torus if the index is 5. In this talk we will provide a generalization of Urbano's Theorem to minimal tori in S^4 by showing that a minimal torus in S^4 has index at least 6 and it is congruent to the Clifford torus if the index is 6. This is a joint work with Prof. Rob Kusner(UMass Amherst).
Fri 03/30 2:30 PM 实验楼105 Chao Xia (Xiamen University) "The Weyl problem in warped product spaces"

Abstract: The Weyl problem studies whether a smooth metric on S^2 with positive Gauss curvature admits a smooth isometric embedding in R^3. The uniqueness part corresponds to Cohn-Vossen's rigidity theorem. The existence part was solved by Nirenberg and Pogorelov independently by using the continuity method. The solution of the Weyl problem is a starting point to well define the Brown-York quasi-local mass. Motivated by the well-definedness of quasi local mass in Schwarzschild manifold, recently, people are interested in study the Weyl problem in warped product spaces. In this talk, I will review the continuity method for this problem and report recent progress made by Li-Wang and Guan-Lu for the Weyl problem in warped product spaces..
The reference:
C. Li and Z. Wang, The Weyl problem in warped product space, arXiv:1603.01350. J. Differential Geom., to appear.
P. Guan and S. Lu, Curvature estimates for immersed hypersurfaces in Riemannian manifolds, Invent. Math. 208 (2017), no. 1, 191-215.
S. Lu, On Weyl's embedding problem in Riemannian manifolds, arXiv:1608.07539.
Fri 04/20 2:30PM 实验楼105 Bennett Chow (University of California, San Diego) "A Survey of Shrinking Gradient Ricci Solitons"

Abstract: We discuss works on shrinking gradient Ricci solitons with an emphasis on some papers of Munteanu and Wang. They have made progress in all dimensions with some stronger results in dimension 4. These objects are interest to Ricci flow because they model finite time singularity formation.
About the speaker: The speaker's homepage at UC San Diego is http://www.math.ucsd.edu/~benchow/.
Fri 05/18 2:30 PM 实验楼105 Victor Ginzburg (University of California, Santa Cruz) "Periodic orbits of Hamiltonian systems: the Conley conjecture and pseudo-rotations"

Abstract: One distinguishing feature of Hamiltonian dynamical systems--a class of systems naturally arising in many physics problems--is that such systems, with very few exceptions, tend to have numerous periodic orbits. In 1984 Conley conjectured that a Hamiltonian diffeomorphism (i.e., the time-one map of a Hamiltonian flow) of a torus has infinitely many periodic orbits. This conjecture was proved by Hingston some twenty years later and similar results for surfaces other than the sphere were established by Franks and Handel. Of course, one can expect the Conley conjecture to hold for a much broader class of phase spaces, and this is indeed the case as has been shown by Gurel, Hein and the speaker. However, the conjecture is known to fail for some, even very simple, phase spaces such as the sphere. These spaces admit Hamiltonian diffeomorphisms with finitely many periodic orbits--the so-called pseudo-rotations--which are of particular interest in dynamics. In this talk, based on the results of Gurel and the speaker, we will examine underlying reasons for the existence of periodic orbits for Hamiltonian systems and discuss the situations where the Conley conjecture does not hold.
About the speaker: The speaker's homepage at UC Santa Cruz is https://ginzburg.math.ucsc.edu/.
Fri 05/18 3:30 PM 实验楼105 Martin Li (The Chinese University of Hong Kong) "Free Boundary Minimal Surfaces in the unit ball"

Abstract: Since the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem, there have been substantial interest in the study of free boundary minimal surfaces in the unit ball. In this talk, we will discuss some very recent results concerning the existence, compactness and rigidity of such objects. We will mention along the way some open questions in this area. Part of these are joint work with A. Fraser; and N. Kapouleas.
Tue 05/29 2:30 PM 数理楼661 Wenxiong Chen (Yeshiva University) "The fractional Laplacian"

Abstract: The fractional Laplacian is a non-local pseudo-differential operator defined by a singular integral. It is quite different from the traditional (local) differential operators. In this talk, we will use simple examples to illustrate the essential differences between the local and nonlocal operators, such as the boundary regularities and Poisson representations. We will show how to construct a super solution to obtain Holder regularity of the solutions on the boundary; we will also show how to construct a sub-solution to prove a Hopf type lemma. If time permitting, we will show the ideas on the proofs of interior regularity (the Schauder estimate).
Fri 06/01 2:30 PM 实验楼105 Lihan Wang (University of Connecticut) "Symplectic Laplacians, boundary conditions and cohomology"

Abstract: Symplectic Laplacians are introduced by Tseng and Yau in 2012, which are related to a system of supersymmetric equations from physics. These Laplacians behave different from usual ones in Rimannian case and Complex case. They contain both 2nd and 4th order operators. In this talk, we will discuss these operators and their relations with cohomologies on compact symplectic manifolds with boundary. For this purpose, we will introduce new boundary conditions for differential forms on symplectic manifolds. Their properties and importance will be discussed.
Tue 06/05 2:30 PM 数理楼661 Jinyu Guo (Xiamen University) "Overdeteminated problems in a ball in Euclidean space"

Abstract: In a celebrated paper "A symmetry problem in potential theory", Serrin initiated the study of elliptic equations under overdetermined boundary conditions. He introduced the moving plane method to prove this problems. In this talk, I mainly talk about Serrin's type overdeteminated problems. Firstly, I will introduce the history background for Serrin's type overdeteminated problems without boundary. Secondly, I will introduce several proof's methods for this kind of problems. Finally, I will talk about our recent results for overdeteminated problems in a ball.
Fri 06/08 2:30 PM 实验楼105 Shaochuang Huang (Tsinghua University) "Harmonic Coordinates, Exhaustion functions and its applications"

Abstract: In this talk, I will prove a harmonic radius estimate and then use it to construct an exhaustion function with bounded gradient and Hessian by Tam's method. Finally, using similar method by F. He and Lee-Tam, I will sketch a proof of short-time existence of Ricci flow.
Fri 06/08 3:30 PM 实验楼105 Man-Chun Lee (The Chinese University of Hong Kong) "Chern Ricci flow on noncompact manifolds and applications"

Abstract: In this work, we study a Hermitian flow of metrics evolving along the Chern Ricci direction. We will discuss a existence criteria of the Chern Ricci flow and hence the Kahler Ricci flow without the assumption of bounded curvature. If time is allowed, I will briefly describe a construction of KRF on non-collapsing manifold with nonnegative bisectional curvature and its application to Yau's uniformization conjecture. This is joint work of Prof. L.F. Tam.
Fri 06/15 2:30 PM 实验楼105 Fang Wang (Shanghai Jiaotong University) "Obata's Rigidity theorem on manifolds with boundary"

Abstract: In this talk, I will introduce some rigidity theorems for the (generalized) Obata equation on manifolds with boundary with different kinds of boundary conditions. Then I will also give two main applications. One application is in the rigidity theorems of Poincare-Einstein manifolds; and the other is in the first eigenvalue problems on manifolds with boundary. This is joint work with Mijia Lai and Xuezhang Chen.
Tue 06/19 3:00 PM 数理楼661 Bingyuan Liu (University of California, Riverside) "Geometric analysis on the Diederich–Fornæss index"

Abstract: In this talk, we discuss the Diederich–Fornæss index in several complex variables. A domain \Omega \subset \mathbb{C}^n is said to be pseudoconvex if -\log(-\delta(z)) is plurisubharmonic in \Omega, where \delta is a signed distance function of \Omega. The Diederich–Fornæss index has been introduced since 1977 as an index to refine the notion of pseudoconvexity. After a brief review of pseudoconvexity, we discuss this index from the point of view of geometric analysis. We will find an equivalent index associated to the boundary of domains and with it, we are able to obtain accurate values of the Diederich–Fornæss index for many types of domains.
Tue 06/26 2:30 PM 实验楼105 Xiaodong Wang (Michigan State University) "From the isoperimetric inequality to Integral inequalities for harmonic functions and holomorphic functions"

Abstract: There are many proofs for the classic isoperimetric inequality. Carleman's proof reduces it to an interesting integral inequality for analytic functions on the unit disc in the plane. As natural generalizations I will discuss some integral inequalities for harmonic functions in higher dimensions. This is based on joint work with Fengbo Hang and Xiaodong Yan. I will also talk about some more recent developments and related inequalities for holomorphic functions in several complex variables if time allows.
Thu 07/05 3:00 PM (NOT the usual time!) 教学楼306 (NOT the usual location!) Boyong Chen (Fudan University) "Weighted Bergman kernel, directional Lelong number and John-Nirenberg exponent"

Abstract: Let $\psi$ be a plurisubharmonic function on the closed unit ball and $K_{t\psi}(z)$ the Bergman kernel on the unit ball with respect to the weight $t\psi$. We show that the boundary behavior of $K_{t\psi}(z)$ is determined by certain directional Lelong numbers of $\psi$ for all $t$ smaller than the John-Nirenberg exponent of $\psi$ associated to certain family of nonisotropic balls, which is always positive.
Note: This is a special lecture from the 2018 Summer school on Finsler geometry.
Thu 07/05 4:00 PM (NOT the usual time!) 教学楼306 (NOT the usual location!) Siqi Fu (Rutgers University) "Estimates of invariant metrics and applications"

Abstract: The Caratheodory and Kobayashi metrics are non-smooth Finsler metrics while the Bergman metric is a Kahler metric on bounded domains in several complex variables. All of them are biholomorphic invariants. In this talk, we will discuss boundary estimates of these invariant metrics and the Bergman kernel. We will also discuss how these estimates can be used to characherize certain geometric properties of the boundary.
Note: This is a special lecture from the 2018 Summer school on Finsler geometry.
Fri 07/06 11:00 AM (NOT the usual time!) 实验楼108 (NOT the usual location!) Meikui Xiong (Northwestern University, Xi'an) "Deformation of canonical metrics in Kahler geometry"

Abstract: In (1), Gabor Szekelyhidi proved a theorem which shows the structure of the deformation space of csck metrics (constant scalar curvature metrics), he made use of the K-stability to obtain his result. Then in (2), Eiji Inoue generalized the result above to the case of Kahler-Ricci solitons. We will survey some theories related to the two results.
(1) Gabor Szekelyhidi, The Kahler-Ricci flow and K-stability, Arxiv:0803.1613.pdf.
(2) Eiji Inoue, The moduli space of Fano manifolds with Kahler-Ricci solitons, Arxiv:1802.08128.pdf.
Sun 07/08 4:00 PM (NOT the usual time!) 教学楼306 (NOT the usual location!) Xiao Zhang (AMSS) "宇宙学中的一些物理与几何问题"

Abstract: 宇宙学原理假设宙在大尺度上是均匀各向同性的，几何上可以用 Robinson-Walker 度规描述。1998年天文观测发现宇宙在加速膨胀，宇宙常数为正。近年来，更精细的测量数据似乎表明宇宙中存在一些特殊性质的区域，显示出有较强的各向异性性质。一些学者发现用Finsler几何可以解释这样的各向异性。本报告将讨论正宇宙常数的正能量定理以及引力波Bondi-Sachs时空的Peeling性质。并探讨这些问题在Finsler几何框架下的可能推广.
Note: This is a special lecture from the 2018 Summer school on Finsler geometry.
Mon 07/09 11:00 AM (NOT the usual time!) 数理楼661 Haojie Chen (Zhejiang Normal University) "Kodaira dimensions of almost complex manifolds"

Abstract: The Kodaira dimension gives a rough classification scheme of complex manifolds up to birational equivalence. It is also introduced on symplectic 4-manifolds and smooth manifolds with dimension less than 4. In this talk, I will present a generalization of Kodaira dimension to almost complex manifolds. I will discuss some structural results including the birational invariance on almost complex 4-manifolds and the relation with symplectic Kodaira dimension. It is in general not a deformation invariant, hence not a diffeomorphism invariant. If time allows, I will discuss some interesting non-integrable almost complex structures with large Kodaira dimension. This talk is based on joint work with Weiyi Zhang.
Mon 07/09 4:00 PM (NOT the usual time!) 教学楼306 (NOT the usual location!) Xiaobo Liu (Peking University) "Integrable System and Moduli Space of Curves"

Abstract: An integrable system consists of mutually commuting flow equations. A well known integrable system is the KdV hierarchy. Integrable systems have deep connections with geometry of moduli spaces of stable curves (i.e. compactifications of moduli spaces of punctured Riemann surfaces). In this talk I will explain how intersection numbers on such moduli spaces provide solutions to the KdV hierarchy. Conjecturally, a variation of such connections might be generalized to Gromov-Witten invariants of smooth projectic varieties.
Note: This is a special lecture from the 2018 Summer school on Finsler geometry.
Tue 07/10 11:00 AM (NOT the usual time!) 数理楼661 Siyuan Lu (Rutgers University) "On a localized Penrose inequality"

Abstract: We consider the boundary behavior of a compact manifold with nonnegative scalar curvature. The boundary consists of two parts: \Sigma_H and \Sigma_O, where \Sigma_H denotes outer minimizing minimal hypersurface. Under suitable assumption on \Sigma_O, we establish a localized Penrose inequality, which can be viewed as a quasi-local version of the Riemannian Penrose inequality. Moreover, in dimension 3, we prove that the equality holds iff it's a domain in Schwarzschild manifold. This is based on joint works with Pengzi Miao.
Tue 07/17 2:30 PM 数理楼661 Yong Wei (Australia National University) "Volume preserving flow and Alexandrov-Fenchel inequalities in hyperbolic space"

Abstract: I will describe my recent work with Ben Andrews and Xuzhong Chen on volume preserving flow and Alexandrov-Fenchel inequalities in hyperbolic space. First, if the initial hypersurface in hyperbolic space has positive sectional curvature, we show that a large class of volume preserving flow preserves the positivity of sectional curvatures, and the flow converges smoothly to a geodesic sphere. This result can be used to show that certain Alexandrov-Fenchel quermassintegral inequalities, known previously for horospherical convex hypersurfaces (by G.Wang and C.Xia (2013)), also hold under the weaker condition of positive sectional curvature. Second, we consider the volume preserving flow of strictly horospherically convex hypersurfaces in hyperbolic space by function of shifted principal curvatures, and apply the convergence result to prove a new class of Alexandrov-Fenchel type inequalities for horospherically convex hypersurfaces.
Wed 07/25 10:30 AM (NOT the usual time!) 数理楼661 Changliang Wang (Mcmaster University) "Linear stability of Riemannian manifolds with Killing spinors"

Abstract: Einstein metrics on a compact manifold are critical points of the normalized total scalar curvature functional. So it is natural to study the behavior of the second variation of the normalized total scalar curvature functional at an Einstein metric. This is known as the linear stability problem of Einstein metrics. In this talk, we will briefly review previous works on this problem, and then I will report our work on the linear stability of some interesting Einstein metrics: Riemannian metrics admitting Killing spinors, and Einstein metrics from the circle bundle construction.

## 2017-2018 1st Semester (2017 Fall)

Schedule Rooms Topics
Tue 09/12 2:00 PM 数理楼661 Organizational meeting.
Tue 09/19 2:30 PM 数理楼661 Siyuan Ma (Albert Einstein Institute) "On Maxwell field and linearized gravity in Kerr spacetime"

Abstract: After the publication of Einstein's theory of General Relativity in 1915, many predictions have been confirmed in the latest one century, culminating at the recent observations of gravitational waves emitted during the merging of binary black holes by LIGO and VIRGO collaborations. Black holes are one of the fundamental predictions, and the one of most interests is the Kerr black hole spacetimes. The metric of a Kerr spacetime describes a rotating, stationary, axisymmetric, asymptotically flat solution to vacuum Einstein equations. One of the most important open problems in mathematical General Relativity is to address the fully nonlinear stability conjecture of Kerr solutions. In this talk, I will present recent results in obtaining energy estimates for both Maxwell field and linearized gravity on Kerr backgrounds, which will advance the field towards this conjecture.
Tue 09/19 3:40 PM 数理楼661 Chao Liu (Monash University) "Cosmological Newtonian limits on large scales"

Abstract: In this talk, I will rigorously answer one basic question in cosmological simulation: on what space and time scales Newtonian cosmological simulations can be trusted to approximate relativistic cosmologies? We resolve this question under a small initial data condition.
Tue 09/26 2:30 PM 数理楼661 Guofang Wang (Freiburg University) "Local Lagrangian embeddings and Hessian surfaces"

Abstract: We will talk about Local Lagrangian embeddings and Hessian surfaces. This is a joint work with Qing Han.
Tue 10/17 2:30 PM 数理楼661 Bo Yang (Xiamen University) "Kahler-Ricci flow on noncompact manifolds (after Huang-Tam and Lee-Tam)"

Abstract: This talk is purely expository. We explain recent works on Kahler-Ricci flow on complete noncompact Kahler manifolds with non collapsed volume and nonnegative bisectional curvature.
Tue 10/24 2:30 PM 数理楼661 Fei He (Xiamen University) "Existence of Ricci flow on noncompact manifolds"

Abstract: This will be a continuation of Bo Yang's talk from last week. We will discuss the short-time existence of Ricci flow on noncompact manifolds with a focus on the recent work of Lee and Tam.
Fri 11/03 2:30 PM 实验楼105 Xi Zhang (University of Science and Technology of China) "Canonical metrics and The Hermitian-Yang-Mills flow on reflexive sheaves"

Abstract: In this talk, we will introduce our recent work on the existence of canonical metrics, Bogomolov type inequalities and the limiting behavior of the Hermitian-Yang-Mills flow on reflexive sheaves. These work are joint with JiaYu Li, YanCi Nie and ChuanJing Zhang.
Fri 11/10 2:30 PM 实验楼105 Bin Zhou (Peking University) "K-energy on polarized compactifications of Lie groups"

Abstract: In this paper, we study Mabuchi’s K-energy on a compactification M of a reductive Lie group G, which is a complexification of its maximal compact subgroup K. We give a criterion for the properness of K-energy on the space of K × K-invariant Kahler potentials. In particular, it turns to give an alternative proof of Delcroix’s theorem for the existence of Kahler-Einstein metrics in case of Fano manifolds M . We also study the existence of minimizers of K-energy for general Kahler classes of M.
Fri 11/10 3:40 PM 实验楼105 Weiming Shen (BICMR, Peking University) "On The Negativity of Ricci Curvatures of Complete Conformal Metrics"

Abstract: A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this talk, I will disscuss whether these metrics have negative Ricci curvatures. We will provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green's function and the positive mass theorem play essential roles in certain cases. On the other hand, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space.
Fri 11/17 3:30 PM 实验楼105 Xiao Zhang (AMSS, Beijing) " The positive energy theorem for asymptotically hyperbolic manifolds"

Abstract: In general relativity, asymptotically hyperbolic manifolds serve as the initial data sets in two cases: (i) asymptotically null infinity for asymptotically flat spacetimes where the cosmological constant is zero; (ii) asymptotically spatial infinity for asymptotically AdS spacetimes where the cosmological constant is negative. The difference is that, in case (i), the second fundamental forms are asymptotic to hyperbolic metrics while in case (ii), the second fundamental forms are asymptotic to zero. We will discuss the positive energy theorem in the two cases. The talk is based on the early work of the speaker as well as the joint work with Wang Yuahua and Xie Naqing.
Fri 11/24 2:30 PM 实验楼105 Zhizhang Wang (Fudan University) "The curvature estimates for convex solutions of some fully nonlinear Hessian type equations"

Abstract: The curvature estimates of quotient curvature equation do not always exist even for convex setting. Thus it is natural question to find other type of elliptic equations possessing curvature estimates. In this paper, we discuss the existence of curvature estimate for fully nonlinear elliptic equations defined by symmetric polynomials, mainly, the linear combination of elementary symmetric polynomials. This is a joint work with Chunhe Li and Changyu Ren.
Fri 11/24 3:40 PM 实验楼105 Naqing Xie (Fudan University) "Toroidal marginally outer trapped surfaces in the closed Friedmann-Lemaitre-Robertson-Walker universe"

Abstract: We explicitly construct toroidal MOTS in the closed FLRW universe. This construction is used to assess the quality of certain isoperimetric inequalities recently proved in axial symmetry. We also show that these constructed toroidal MOTS are unstable. This talk is based on a joint work with Patryk Mach.
Fri 12/01 2:30 PM 实验楼105 Frederick Tsz-Ho Fong (Hong Kong University of Science and Technology) "Rigidity of Self-Expanders of Inverse Curvature Flows"

Abstract: In this talk, the speaker will investigate a large class of curvature flows by degree -1 homogeneous functions of principal curvatures in Euclidean spaces. This class curvature flows include the well-known inverse mean curvature flow and many others in the current literature. Self-expanding solutions to these curvature flows are solutions that expanding homothetically without changing their shapes. We will talk about uniqueness, rigidity, and constructions of both compact and non-compact self-expanding solutions to these flows. Part of these are joint work with G. Drugan, H. Lee; P. McGrath; and A. Chow, K. Chow.
Fri 12/08 3:30 PM 实验楼105 Hui Ma (Tsinghua University) "Uniqueness of closed self-similar solutions to $\sigma_k^{\alpha}$-curvature flow"

Abstract: In this talk we will show the uniqueness of closed self-similar solutions to $\sigma_k^{\alpha}$-curvature flow. It is based on the joint work with Shanze Gao and Haizhong Li.
Fri 12/22 3:00 PM 实验楼105 Daniel Zhuangdan Guan (UC Riverside) "Recent progress on compact Kaehler-Einstein manifolds with cohomogeneity one metrics"

Abstract: Although there are many known K\"ahler-Einstein manifolds, there is so far no very practical method to check a given compact Fano manifold to be K\"ahler-Einstein or not.This is also true for the K\"ahler metrics with constant scalar curvatures or Calabi extremal metrics. The situation for a cohomogeneity one metrics was completely resolved.For the type III case, it was solved in my dissertation in 1992. For the remain type I and II case, the existence is equivalent to the negativity of a topological integral.The type I case was published in 2011. Therefore, the problem is reduced to check the negativity for classes of Fano manifolds. Recently, we use computer to get some insight into a class of type I Fano manifolds. This work is a joint work with a group of students. .
Wed 12/27 3:00 PM 行政楼313 Daniel Zhuangdan Guan (UC Riverside) "Recent Progress in the classification of complex homogeneous spaces"

Abstract: A manifold M is a homogeneous space if M=G/H with G a finite dimensional group and H a closed subgroup. M is a complex homogeneous space if J is the given complex structure on M such that J is invariant under the action of G. Homogeneous space is a classical area of differential geometry.The most famous work was the classification of real (and complex) semi-simple Lie groups and the symmetric spaces. The K\"ahler homogeneous space was classified by Dorfmeister and Nakajima in 1988. The pseudo-k\"ahler homogeneous space with reductive G was classified by Dorfmeister and Guan in 1989. In the general compact complex homogeneous case, the classification reduced to the parallelizable case, i.e., in which H is discrete. In the late 1990's we proved that if G/H is a compact complex parallelizable manifold,then the semi-simple part of G is locally a product of complex simple Lie group of type A.A classification of the compact complex homogeneous space with an invariant volume was also finally classified.A complete classification of the compact complex space with a pseudo-k\"ahler structure (non-necessary invariant) was given in 2007. Recently, compact complex homogeneous space with an invariant locally conformal K\"ahler structure was classified and similarly for the cohomogeneity one case.
Fri 12/29 2:30 PM 实验楼105 Yunhui Wu (Tsinghua University) "The Weil-Petersson geometry of the moduli of curves for large genus"

Abstract: We study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniform Lipschitz on the Teichmuller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of the moduli space of Riemann surfaces of genus $g$ with $n$ punctures as a function of $g$ and $n$. We show that the Weil-Petersson inradius is comparable to $\sqrt{\ln{g}}$ with respect to $g$, and is comparable to $1$ with respect to $n$.
Fri 12/29 3:30 PM 实验楼105 Qing Han (University of Notre Dame) "Nonexistence of Poincare-Einstein Fillings on Spin Manifolds"

Abstract: In this talk, we discuss whether a conformal class on the boundary M of a given compact manifold X can be the conformal infinity of a Poincare-Einstein metric in X. We construct an invariant of conformal classes on the boundary M of a compact spin manifold X of dimension 4k with the help of the Dirac operator. We prove that a conformal class cannot be the conformal infinity of a Poincare-Einstein metric if this invariant is not zero. Furthermore, we will prove this invariant can attain values of infinitely many integers if one invariant is not zero on the above given spin manifold. This talk is based on a joint work with Gursky and Stolz.
Tue 01/09 2:30 PM 数理楼661 Guohuan Qiu (McGill University) "Interior Hessian estimates for sigma-2 equations in dimension three"

Abstract: The interior regularity for solutions of the sigma_2 Hessian equation is a longstanding problem.Heinz first derived this interior estimate in dimension two. For higher dimensional Monge-Ampere equations, Pogorelov constructed his famous counter-examples even for f constant and convex solutions. Caffarelli-Nirenberg-Spruck studied more general fully nonlinear equations such as \sigma_{k} equations in their seminal work. And Urbas also constructed counter-examples with k greater than 3. The only unknown case is k=2. A major breakthrough was made by Warren-Yuan, they obtained a prior interior Hessian estimate for the equation \sigma_2=1 in dimension three.In this talk, I will present my recent work on how to deal this problem for a more general case in dimension three.
Fri 01/12 3:30 PM 实验楼105 Xingwang Xu (Nanjing University) "Q and R"

Abstract: In this talk, I should focus on the conformal invariant equations of higher order. We interpolate them in terms of conformal geometry. Natural geometric information provides the maximum principle for such equations. This is a joint work with Mr. Weixi Wang.
Fri 01/12 4:30 PM 实验楼105 Robert Kusner (University of Massachusetts at Amherst) "Coplanar CMC surfaces, complex projective structures, and polynomial quadratic differentials"

Abstract: Complete embedded constant mean curvature (CMC) surfaces of fixed, finite topology form a finite-dimensional moduli space. This moduli space is a real-analytic variety parametrized by the asymptotic data of the surfaces, and possibly by some square-integrable Jacobi fields. For coplanar CMC surfaces of genus 0 with k ends, such Jacobi fields must vanish, and this moduli space can be explicitly described: it is diffeomorphic to the space of k-point spherical metrics; these can be described, in turn, by holomorphic immersions from the plane to the 2-sphere whose Schwarzian is a polynomial with degree depending on k. The CMC surfaces corresponding to the polynomials 0 and 1 are, respectively, the round sphere and the 1-parameter family of unduloids, while those which correspond to the polynomial z are the 3-parameter family of triunduloids. Byviewing the Schwarzian as a quadratic differential and its real foliations, a compelling picture of this correspondence emerges. (If time permits, a new construction of coplanar CMC surfaces with genus 1, all of whose ends are cylindrical, will also be described.).
Thu 01/18 2:30 PM 实验楼108 Miaomiao Zhu (Shanghai Jiaotong University) "Existence of solutions of a boundary value problem for Dirac-harmonic maps"

Abstract: In this talk, we shall present some recent progresses on the heat flow approach to the existence of solutions of a boundary value problem for Dirac-harmonic maps. These are joint works with Jurgen Jost and Lei Liu.

Last modified: 12/27/2017 by the geometric analysis group at XMU Math.