Workshop on Perspectives in Geometric Analysis (Part A)
Speaker:
Sponsored by:
Time: From 2016-07-01 08:00 To 2016-07-03 18:00
Venue: Lecture Hall, Jiayibing Building, BICMR
Confirmed speakers:
Yann Bernard,Monash
Jingyi Chen,Univ. British Columbia
Tamas Darvas,University of Maryland
Fuquan Fang,Capital Normal University
Hao Fang,University of Iowa
Wenshuai Jiang,Peking University
Xishen Jin,University of Science and Technology of China
Tobias Lamm,Karlsruhe Institute of Technology
Yuxiang Li,Tsing Hua Univercity
Gang Liu,Berkeley
Chao Qian,Beijing Institute of Technology
Liangmin Shen,Univ. British Columbia
Takashi Shioya,Tohoku
Rivière Tristan,ETH Zurich
Lu Wang,University of Wisconsin-Madison
Haotian Wu,Oregon University
Jie Wu,Zhejiang University
Chengjie Yu,Shantou University
Jiazu Zhou,Southwest University
Lei Zhang,University of Florida
Qi Zhang,UC Riverside
Xi Zhang,University of Science and Technology of China
Zhou Zhang,University of Sydney
Title and abstract:
Claudio Arezzo (ICTP)
Title: Kahler constant scalar curvature metrics on blow ups and resolutions of singularities
Abstract: In this seminar I will present a gluing construction for Kahler constant scalar curvature and extremal metrics starting from a singular compact or ALE spaces with isolated singularities of orbifold or conical type.
Joint work(s) with A. Della Vedova, R. Lena, L. Mazzieri and C. Spotti.
Yann Bernard (Monash U, Australia)
Title: Ends of immersed minimal and Willmore surfaces in asymptotically flat spaces
Abstract: We study ends of an oriented, immersed, non-compact, complete Willmore surfaces, which are critical points of the integral of the square of the mean curvature, in asymptotically flat spaces of any dimension; assuming the surface has $L^2$-bounded second fundamental form and satisfies a weak power growth on the area. We give the precise asymptotic behavior of an end of such a surface. This asymptotic information is very much dependent on the way the ambient metric decays to the Euclidean one. Our results apply in particular to minimal surfaces. This is joint-work with Tristan Rivière.
Jingyi Chen (Univ. British Columbia)
Title: Geometry of Lagrangian self-shrinking tori
Abstract: We will discuss finiteness of the entropy values of Lagrangian self-shrinking tori in the complex plane under a given upper bound and applications to construct piecewise Lagrangian mean curvature flow for torus. This is joint work with John Ma.
Tamas Darvas (University of Maryland)
Title: Metric geometry on the space of Kahler metrics
Abstract: We introduce the $L^p$-Mabuchi Finsler structure on the space of Kahler metrics. We argue that the completion of the associated path length metric structures are well known objects from finite energy pluripotential theory. Time permitting we will mention how the use of the arising metric geometry is useful in settling questions about the long time behavior of the Calabi flow and J-properness.
Fuquan Fang (Capital Normal University)
Title: Reflections in Riemannian geometry
Abstract: We provide an equivariant description / classification of all com-plete (compact or not) nonnegatively curved manifolds M together with a co-compact action by a reflection group W, andmoreover, classify such W. In particular, we show that the building blocks consist of the classical constant curvature models and generalized open books with nonnegatively curved bundle pages, and derive a corresponding splitting theorem for the universal cover. This is a joint work with Karsten Grove.
Hao Fang(University of Iowa)
Title: Volume bounds for conic 2-spheres with curvature bounds
Abstract: We discuss the best volume bounds for conic 2-spheres with curvature bounds. In particular, we compute the Gromov volume for conic 2-spheres. We show that extremal metrics have rotational symmetry. Also, we will discuss some technical details dealing with negative curvature case. This is a joint work with Mijia Lai.
Wenshuai Jiang (Peking University)
Title: L^2 curvature bounds on manifolds with bounded Ricci curvature
Abstract: In this talk, we will discuss the L^2 curvature estimates on manifolds with bounded Ricci curvature and noncollapsing volume. Firstly, we will talk about the background, then we introduce the concept of neck region which appears everywhere in our proof. After that, we would sketch the whole proofs. At last, we would focus on the technical details and some new observations on neck region. This is joint work with Prof. Aaron Naber.
Xishen Jin (University of Science and Technology of China)
Title: Twisted and conical K\"ahler-Ricci soliton on Fano manifold
Abstract: In this talk, we will show the relation between properness of energy functional and existence of twisted K\"ahler-Ricci soliton. Furthermore, the existence of conical K\"ahler-Ricci soliton will be considered. In particular, under some assumptions, we get the properness of modified log K-energy and the existence of conical K\"ahler-Ricci soliton with suitable cone angle. This work is jointed with Jiawei Liu and Xi Zhang.
Tobias Lamm(Karlsruhe Institute of Technology)
Title: Conformal Willmore tori in \R^4“
Abstract: In this talk I am going to present recent existence and non-existence results for conformal Willmore Tori in $\mathbb{R}^4$ which were obtained in a collaboration with Reiner M. Schätzle (Tübingen).
Yuxiang Li (Tsing Hua Univercity)
Title: Willmore minimizers when prescribed isoperimetric ratio goes to 0
Abstract: Let $f_\sigma:S^2\rightarrow R^3$ be embedding which attains $$\beta_\sigma=\inf_{\mu(f)=1,V(f)=\sigma}W(f).$$ It was proved by Schygulla that the image of $f_\sigma$ converges to a double sphere in the sense of varifold as $\sigma\rightarrow 0$.
In this talk, we will show that $f_\sigma$ converges to a double sphere and a catenoid in the bubble tree sense. This is a joint work with E. Kuwert.
Gang Liu (University of California, Berkeley)
Title: Gromov-Hausdorff limit of Kahler manifolds with bisectional curvature lower bound
Abstract: We prove that the Gromov-Hausdorff limit of Kahler manifolds with bisectional curvature lower bound and noncollapsed volume is homeomorphic to a normal complex analytic space. As a consequence, if M is a complete noncompact Kahler surface with positive bisectional curvature and noncollapsed volume, then it is simply connected.
Chao Qian (Beijing Institute of Technology)
Title: Isoparametric foliation and its applications on related geometric problems
Abstract: In this talk, we will discuss two sequences of minimal isoparametric hypersurfaces, constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres.
Eells and Lemaire [EL83] posed a problem to characterize the compact Riemannian manifold M for which there is an eigenmap from M to S^n. As another application of our constructions, the focal maps give rise to many examples of eigenmaps from minimal isoparametric hypersurfaces to unit spheres.
Most importantly, by investigating the second fundamental forms of focal submanifolds of isoparametric hypersurfaces in unit spheres, we provide infinitely many counterexamples to two conjectures of Leung [Le91] (posed in 1991) on minimal submanifolds in unit spheres.
Notice that these conjectures of Leung have been proved in the case that the normal connection is flat.
This talk is based on the joint work with Prof. Zizhou Tang.
Liangming Shen(Univ. British Columbia)
Title: Conical Kahler-Einstein metrics along a simple normal crossing divisor
Abstract: On a Fano Kahler-Einstein manifold with a simple normal crossing divisor which is proportional to c_{1}(M), if the divisor satisfies certain conditions, we prove that there exist conical Kahler-Einstein metrics with prescribed cone angles along the divisor. If time permits, we will introduce a new curvature estimate for conic metrics along one simple normal crossing divisor.
Takashi Shioya (Tohoku University)
Title: High-dimensional spaces in metric measure geometry
Abstract: Gromov introduced a new topology on the space of metric measure spaces, which is weaker than the measured Gromov-Hausdorff topology and is based on concentration of measure phenomenon.
In this talk, we show some of our works on convergence of spaces with unbounded dimension in Gromov’s topology.
Rivière Tristan(ETH Zurich)
Title: Willmore Minmax Surfaces and the Cost of the Sphere Eversion
Abstract: We develop a general Minmax
procedure in Euclidian spaces for constructing Willmore surfaces of non zero
indices. We implement this procedure to the Willmore Minmax Sphere Eversion in
the 3 dimensional euclidian space in order to compute the cost of this famous
eversion.
Lu Wang (University of Wisconsin-Madison)
Title: Hypersurfaces of Low Entropy
Abstract: The entropy is a natural geometric functional introduced by Colding-Minicozzi to study the singularities of mean curvature flow, and it roughly measures the complexity of a hypersurface of Euclidean space. In this talk, I will survey some recent progress with Jacob Bernstein on understanding the geometry and topology of hypersurfaces with low entropy.
Haotian Wu (University of Oregon)
Title: Asymptotic shapes of neckpinch singularities in geometric flows.
Abstract: Geometric flows such as Ricci flow and mean curvature flow are natural and important tools to understand the geometry and topology of Riemannian manifolds. Geometric flows are nonlinear parabolic (heat) partial differential equations that tend to develop singularities in finite time. A useful approach to analyzing the singularities is the technique of matched asymptotics, which can provide detailed and precise information including the rates of curvature blow-up, the set of points where a singularity forms, and the behavior of the solution in a space-time neighborhood of that singularity. In this talk, we will survey the results concerning the asymptotic shapes of neckpinch singularities in Ricci flow and mean curvature flow.
Jie Wu (Zhejiang University)
Title: Geometric inequalities for hypersurface in H^n
Abstract: In this talk, by using the inverse curvature flow we first establish an optimal Sobolev type inequality for hypersurfaces in H^n. As an application, we obtain Alexandrov-Fenchel inequalities for curvature integrals. Secondly, I will talk about the Alexandrov-Fenchel inequalities with weight in H^n, which is related to the recent study of the Penrose inequality for various mass. This is a joint work with Yuxin Ge and Guofang Wang.
Chengjie Yu (Shantou Univeristy)
Title: Estimate of Higher Steklov Eigenvalues
Abstract:In this talk, we will first give a brief survey of the estimates of Steklov eigenvalue. Then, some new estimates for higher Steklov eigenvalues will be discussed.
Jiazu Zhou(Southwest University)
Title: Isoperimetric problems and Alexandrov-Fenchel inequalities
Abstract: The classical isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length. The classical isoperimetric problem was known in the Ancient Greece. However, the first mathematically rigorous proof was obtained only in the 19th century. The classical Brunn-Minkowski theory, also known as the theory of mixed volumes, originated with Minkowski when he combined his concept of mixed volume with the Brunn-Minkowski inequality. One of Minkowski's major contributions to the theory was to show how his theory could be developed from a few basic concepts, such as support function, vector addition, and volume.
The isoperimetric problem was characterized by isoperimetric inequality. The natural extensions are Minkowski inequalities and Alexandrov-Fenchel inequalities for mixed volumes. Minkowski inequality in a normed space is equivalent to Sobolev inequality. The past investigation found that those known geometric inequalities are equivalent to some analytic inequalities. The latest research focus on finding new (Bonnesen-style) isoperimetric imequalities and Alexandrov-Fenchel inequalities that have analogues in differential geometry, complex geometry, algebraic geometry and convex geometric analysis.
Recent progresses towards an Orlicz-Brunn-Minkowski theory was made by
Lutwak-Yang-Zhang (Adv. Math. 223 (2010) 220-242, J. Diff. Geom. 84 (2010) 365
- 387) and Ludwig-Reitzner (Adv. Math.
224 (2010) 2346-2360, Ann. of Math. 172 (2010) 1223-1271) for valuation, and
the dual Orlicz-Brunn-Minkowski theory was made by Xu-Zhou-Zhu (Adv.Math. 264(2014)
700-725) and Gardner-Hug-Weil-Ye (JMAA, 430 (2015), no. 2, 810-829).
Lei Zhang(University of Florida)
Title: Local Mass Concentration and A priori estimate for singular Toda systems of Rank 2.
Abstract: A Toda system is a nonlinear second order elliptic system with exponential nonlinearity. It is very commonly observed in physics and has many ties with algebraic geometry. From analytic viewpoints it is challenging since the solutions do not have symmetry, maximum principles cannot be applied and the structures of global solutions are incredibly complicated. In this joint work with Chang-shou Lin and Juncheng Wei, we use a unified approach to discuss all rank two singular Toda systems. First for local systems we prove that all weak limits of mass concentration belong to a very small finite set. Then for systems defined on compact Riemann surface we establish a priori estimate. Our approach is a combination of delicate blowup analysis and fundamental tools from algebraic geometry.
Qi Zhang (UC Riverside)
Title: New Volume Comparison results and Applications to degeneration of Riemannian metrics.
Abstract:We consider a condition on the Ricci curvature involving vector fields, which is broader than the Bakry-\'Emery Ricci condition. Under this condition a volume comparison, Laplacian comparison, isoperimetric inequality and gradient bounds are proven on the manifold.
Specializing to the Bakry-\'Emery Ricci curvature condition, we initiate an approach to work on the original manifold, which yields, under a weaker than usual assumption, the results mentioned above for the original manifold. These results are different from well known ones in the literature where the conclusions are made on the weighted manifold instead.
Applications on convergence and degeneration of Riemannian metrics under this curvature condition are given.
To this effect, the gradient of the potential function is allowed to have singularity of order close to $1$ while the traditional method of weighted manifolds allows bounded gradients. The result also covers general solitons rather than just gradient solitons.
This is a joint work with Zhu, Meng.
Xi Zhang ( University of Science and Technology of China)
Title: The Hermitian-Yang-Mills flow on reflexive Higgs sheaves
Abstract:In this talk, we consider the asymptotic behavior of the Hermitian-Yang-Mills flow on reflexive Higgs sheaves. We prove that if a reflexive Higgs sheaf is not stable, then it breaks up into a direct sum of Hermitian-Einstein reflexive Higgs sheaves via the Hermitian-Yang-Mills flow. Furthermore, we show that the direct sum is isomorphic to the double dual of the graded Higgs sheaves associate to Harder-Narasimhan-Seshadri filtration. This work is joint with JiaYu Li and ChuanJing Zhang.
Zhang Zhou (The University of Sydney)
Title: Mean Curvature Flow over Almost Fuchsian Manifolds
Abstract: we consider mean curvature flow in the non-Euclidean setting. More specifically, the ambient 3-manifold is almost Fuchsian, which is hyperbolic and of great interest in low-dimension topology and differential geometry. The (closed) surface of evolution under study is a graph. After some general discussion, we restrict ourselves to the case of Fuchsian manifold. Under a mild assumption on the graph property of the initial surface, we can justify the smooth convergence to the minimal surface at time infinity. The discussion could also be helpful in constructing examples of finite time singularities. This is a joint work with Zheng Huang (CUNY) and Longzhi Lin(UCSC).