Lecture Room: New Math Building 308

Tentative Schedule

Schedule
Wednesday, December  10
Time
9:30a  ~ 10:00a	Registration
10:00a~10:10a	Opening Remarks
10:10a~11:00a	Richard Schoen (Stanford University, USA) Title: The geometry of static configurations in relativity
11:00a~11:10a	Break
11:10a ~12:00	Luen Fai Tam (Chinese University of Hong Kong, China) Title: Boundary behaviors of compact manifolds and scalar curvature
12:00 ~2:00p	Reception
2:00p ~2:30p	Takashi Okayasu (Ibaraki University, Japan) Title: A construction of complete hypersurfaces with constant scalar curvature in the Euclidean space
2:30p ~ 3:00p	Makoto Narita (Okinawa National College of Technology, Japan) Title: On spherically symmetric gravitational collapse in the Einstein-Gauss-Bonnet theory
3:00p ~ 3.20p	Break
3:20p ~ 4:10p	Naichung Conan Leung(The Chinese University of Hong Kong, China) Title: Compactness for the massive Seiberg-Witten equation
Thursday, December  11
9:30a~10:20a	Jaigyoung Choe (Korea Institute for Advanced Study, Korea) Title: First eigenvalue of the Laplacian on minimal surfaces in S3
10:20a~10:40a	Break
10:40a~11:30a	Chiung-Jue Anna Sung (National Tsing Hua University, Taiwan) Title: The upper bound estimate of eigenvalue on surface
11:30a~11:40a	Break
11:40a~12:10	Photograph
12:10 ~ 1:30p	Lunch
1:30p ~ 2:20p	Kazuo Akutagawa (Tokyo University of Science, Japan) Title: The uncertainty principle lemma under gravity and the discrete spectrum of  Schrödinger operators
2:20p ~ 2.40p	Break
2:40p ~ 3:30p	Zhiqin Lu (UC Irvine, USA) Title: Normal Scalar Curvature Conjecture and its applications
3:30p ~ 3.40p	Break
3:40p~4:10p	Yuan-Jen Chiang (University of Mary Washington, USA) Title: Biwave Maps
4:10p~4:40p	Rung-Tzung Huang (Korea Institute for Advanced Study, Korea) Title: Refined analytic torsion for manifolds with boundary
5:30p	Dinner
Friday, December  12
9:30a~10:20a	Takao Yamaguchi (Tsukuba University, Japan) Title: Collapsing and essential covering of Riemannian manifolds
10:20a~10:40a	Break
10:40a~11:30a	Xi-Ping Zhu (Sun Yat-Sen University, China) Title: Classification of compact four-manifolds with positive isotropic curvature
11:30a~11:40a	Break
11:40a~12:10	Takumi Yokota (University of Tsukuba, Japan) Title: Perelman's reduced volume and gap theorem for the Ricci flow
12:10a~13:30	Lunch
	Break Points of interest: Yangmingshan National Park Taipei 101 National Palace Museum Beitou Hot Springs Region (related links Brief history of Beitou, Xinbeitou hotspring and Beitou dinstinguishing culture) Yingge Sansia Old Street Danshuei Hongmao Castle Official Link: Yangmingshan National Park Taipei 101 National Palace Museum Sansia Old Street Danshuei Hongmao Castle
Saturday, December  13
9:30a~10:20a	Robert Bartnik (Monash University, Australia) Title: Second Variation in Relativity
10:20a~10:40a	Break
10:40a~11:30a	Mu-Tao Wang (Columbia University, USA) Title: Isometric embeddings of surfaces and quasilocal gravitational energy
11:30a~11:40a	Break
11:40a~12:10	Lan-Hsuan Huang (Stanford University, USA) Title: Constant Mean Curvature Foliation for Isolated Systems
12:10 ~ 1:30p	Lunch
1:30p ~ 2:20p	Yng-Ing Lee (National Taiwan University, Taiwan) Title: Self-similar solutions and translating solutions for Lagrangian mean curvature flow
2:20p ~ 2.40p	Break
2:40p ~ 3:30p	Mark Haskins (Imperial College London,  UK) Title: Singular special Lagrangian n-folds
3:30p ~ 3.40p	Break
3:40p~4:10p	Mei-Lin Yau (National Central University, Taiwan) Title: Monodromy groups of Lagrangian tori in the symplectic 4-space
Sunday, December  14
9:30a~10:20a	Akito Futaki (Tokyo Institute of Technology,  Japan) Title: Hilbert series and obstructions to asymptotic semistability
10:20a~10:40a	Break
10:40a~11:30a	Jih-Hsin Cheng (Academic Sinica, Taiwan) Title: The positive mass problem in CR geometry
11:30a~11:40a	Break
11:40a~12:10	Tee-How Loo (University of Malaya, Malaysia) Title: Real Hypersurfaces in a Complex Space Form with h-parallel Shape Operator
12:10 ~ 1:30p	Lunch
1:30p ~ 2:20p	Martin Guest (Tokyo Metropolitan University,  Japan) Title: Differential geometric aspects of quantum cohomology
2:20p ~ 2.40p	Break
2:40p ~ 3:30p	Chin-Lung Wang(National Taiwan University, Taiwan) Title: Singular mean field equations on tori

Title and Abstract  (Invited Talks)

Kazuo Akutagawa (Tokyo University of Science, Japan) Title: The uncertainty principle lemma under gravity and the discrete spectrum of  Schrödinger operators Abstract: This is a joint work with Hironori Kumura (Shizuoka Univ., Japan). In this talk, we will discuss the finiteness and infiniteness of the discrete spectrum for a Schrödinger operator - Dg + V on a complete noncompact n-manifold (Mn, g) (n ³ 3). When (Mn, g) = Rn, it is well known that the borderline behavior of the potential V(x) is |x|-2 for large |x|, and that Reed-Simon and Kirsch-Simon gave a complete answer to the question whether - D+ V on Rn has an infinite number of the discrete spectrum. In the proof, the uncertainty principle lemma for the Laplacian D on Rn is important. We will give a generalization of the uncertainty principle lemma on Rn to that on a complete noncompact manifold with a pole. Replacing Rn by some classes of complete noncompact manifolds (not necessarily with a pole), we also establish a version of the above criterion.

Robert Bartnik (Monash University, Australia) Title: Second Variation in Relativity Abstract: Formulae for the first and second "variations" (ie. derivatives) have been widely used in the case of Riemannian manifolds. Their counterparts in the Lorentzian case also have had some remarkable successes, some of which I will describe.

Jih-Hsin Cheng (Academic Sinica, Taiwan) Title: The positive mass problem in CR geometry Abstract: I'll introduce asymptotically Heisenberg (flat pseudohermitian) manifolds on which the mass is defined. We consider the CR Laplacian L on a closed CR manifold. We identify the mass with the first nontrivial coefficient in the Green function expansion of L. We deduce an integral formula for the mass. We discuss the positivity of the mass. As an application, we can solve the Yamabe minimizer problem in CR geometry under some reasonable conditions.

Jaigyoung Choe (Korea Institute for Advanced Study, Korea) Title: First eigenvalue of the Laplacian on minimal surfaces in S3 Abstract: It will be shown that the first eigenvalue of the Laplacian equals 2 on the compact embedded minimal surfaces in S3 which are constructed by Lawson, Karcher-Pinkall-Sterling, and Kapouleas-Yang.

Akito Futaki (Tokyo Institute of Technology,  Japan) Title: Hilbert series and obstructions to asymptotic semistability Abstract: Given a polarized manifold there are obstructions for asymptotic semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature Kähler (cscK) metric. A natural question is whether or not the other obstructions are linearly dependent on the obstruction to the existence of a cscK metric. The purpose of this paper is to see that this is not the case by exhibiting toric Fano threefolds in which these obstructions span at least two dimension. To see this we show that on toric Fano manifolds these obstructions are obtained as derivatives of the Hilbert series. This last observation may be regarded as an extension of the volume minimization of Martelli, Sparks and Yau.

Martin Guest (Tokyo Metropolitan University,  Japan) Title: Differential geometric aspects of quantum cohomology Abstract: The quantum differential equations are a certain overdetermined system of linear p.d.e. associated to quantum cohomology. They appear in widely differing contexts: algebraic and symplectic geometry, singularity theory, number theory, differential geometry (and string theory, where they were discovered).  In this talk shall focus on differential geometry: the quantum cohomology of a manifold corresponds to a certain pluriharmonic map. In particular the quantum cohomology of the 2-sphere corresponds to a certain surface of constant mean curvature in Minkowski space.

Mark Haskins (Imperial College London,  UK) Title: Singular special Lagrangian n-folds Abstract: We discuss recent progress on understanding singular special Lagrangian n-folds. Our focus will be on joint work with N. Kapouleas using gluing methods to construct a wide variety of special Lagrangian cones in every dimension three and greater.

Yng-Ing Lee (National Taiwan University, Taiwan) Title:Self-similar solutions and translating solutions for Lagrangian mean curvature flow Abstract: Singularities in mean curvature flow often models on soliton solutions, which are moved by rescaling or translation by mean curvature flow. Hence it is an important issue to understand these solutions. In this talk, I will report a recent work with D. Joyce and M.P. Tsui where families of new self- similar solutions and translating solutions for Lagrangian mean curvature flow are constructed. These examples appear to have some unexpected properties, which include self-expanders and translating solutions with arbitrarily small Lagrangian angle. Our translating solutions are constructed from self-similar solutions. This is a new approach and can be applied to other cases.

Naichung Conan Leung (The Chinese University of Hong Kong, China) Title:Compactness for the massive Seiberg-Witten equation Abstract: In this talk, I will explain my joint work with Xu Ming. I will introduce the massive Seiberg-Witten equation by allowing the spinor field to have energy up to any fixed level. The main result is the compactness of its moduli space. To prove the compactness theorem, we use the eigenvalue estimate by Vafa and Witten; a repeated use of the Weitzenbock formula; a control of individual eigencomponent of the spinor field and bootstrapping arguments. I will also discuss our ongoing project on using this moduli space to define new invariants for four manifolds.

Zhiqin Lu (UC Irvine, USA) Title:Normal Scalar Curvature Conjecture and its applications Abstract: In this talk, we show how to prove the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture. As an application, we will use our new results to re-exam the classical pinching theorems of minimal submanifolds in spheres. Better pinching theorems are obtained.

Richard Schoen (Stanford University, USA) Title:The geometry of static configurations in relativity Abstract: In this lecture we will discuss progress and problems related to the static n-body problem in relativity. It has been known for some time that static black holes are rotationally symmetric and thus isometric with the Schwarzschild solution. It has been shown recently that static perfect fluids are rotationally symmetric. Such results are not true when one considers more general static bodies. On the other hand one expects that such solutions satisfy some geometric restrictions. We will describe this problem and present a recent result which restricts the geometry of static solutions.

Chiung-Jue Anna Sung (National Tsing Hua University, Taiwan) Title: The upper bound estimate of eigenvalue on surface Abstract: In this talk, we will describe a general result concerning the upper bound estimate of eigenvalues on surfaces.

Luen Fai Tam (The Chinese University of Hong Kong, China) Title: Boundary behaviors of compact manifolds and scalar curvature Abstract: In this talk, we will discuss how to use quasi spherical metrics and positive mass theorems to study the boundary behavior of a compact manifold in terms of the lower bound of the scalar curvature of the manifold, the mean curvature and the Gaussian curvature of the boundary. The relation of the results and quasi-local mass in general relativity will also be discussed.

Chin-Lung Wang (National Taiwan University, Taiwan) Title: Singular mean field equations on tori Abstract: We show that the Green functions on flat tori can have either 3 or 5 critical points only. There does not seem to be any direct method to attack this problem. Instead, we have to employ certain singular mean field equations (MFE) to study it. The distribution of critical points over the moduli of tori gives rise to a new kind of geometry which is fundamental to more general MFE's. The purpose of this talk is to study this correspondence between critical points and solutions of MFE's via the theory of elliptic functions.

Mu-Tao Wang (Columbia University, USA) Title: Isometric embeddings of surfaces and quasilocal gravitational energy Abstract: In general relativity, the gravitational field is represented by the Lorentz metric of spacetime. Einstein's field equation relates the gravitation and matter fields. The total energy contained in a bounded region in the universe has contributions from both sources. Matter fields have energy density and the energy can be evaluated as a flux integral over the boundary. However, the equivalence principle prevents the existence of energy density for gravitation. On a large scale, the gravitation dominates but the measurement of the gravitational energy is extremely subtle as it depends of the geometric configuration which is distorted by the underlying presumably nonflat Lorentz metric. In this talk, I shall explain a new way to measure the total energy contained in a bounded region using tools from differential geometry and PDE's, in particular the isometric embedding of the boundary surface. This is a joint work with S. T. Yau at Harvard.

Takao Yamaguchi (Tsukuba University, Japan) Title: Collapsing and essential covering of Riemannian manifolds Abstract:In this talk I will introduce the notion of an essential covering of a Riemannian manifold , and give a uniform bound of the number of metric balls in the essential covering in the framework of collapsing

Xi-Ping Zhu (Sun Yat-Sen University, China) Title: Classification of compact four-manifolds with positive isotropic curvature Abstract: In this talk, I report our recent work on the classification of compact four-manifolds with positive isotropic curvature. We proved that a compact four-manifold admitting a metric of positive isotropic curvature if and only if it is diffeomorphic to S4, or RP4, or quotients of S3xR by a cocompact fixed point free subgroup of the isometry group of the standard metric of S3xR , or a connected sum of them. In particular, this gives an affirmative answer to a conjecture of R. Schoen in dimension four.

Title and Abstract  (Contributed Talks)

Yuan-Jen Chiang  (University of Mary Washington, USA) Title: Biwave Maps Abstract: We generalize wave maps to biwave maps. We prove that the composition of a biwave map and a totally geodesic map is a biwave map. We give examples of biwave non-wave maps. We show that if f is a biwave map into a Riemannian manifold under certain circumstance, then f is a wave map. We verify that if f is a stable biwave map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, then f is a wave map. We finally obtain a theorem involving an unstable biwave map.

Lan-Hsuan Huang  (Stanford University, USA) Title: Constant Mean Curvature Foliation for Isolated Systems Abstract: We will discuss the existence and the uniqueness of the foliation by  stable spheres with constant mean curvature for asymptotically flat  manifolds satisfying the Regge-Teitelboim condition at infinity.  This  work generalizes the earlier results of Huisken-Yau, Ye, and Metzger.  We will also discuss the concept of the center of mass in  general  relativity.

Rung-Tzung Huang  (Korea Institute for Advanced Study, Korea) Title: Refined analytic torsion for manifolds with boundary Abstract: The refined analytic torsion was introduced by M. Braverman and T. Kappeler as a canonical refinement of analytic torsion on odd dimensional closed manifolds. In this talk I will describe a new boundary condition, by using the Hodge decomposition and the symplectic structure of the space of the complex flat bundle valued differential forms on the boundary of an odd dimensional compact manifold, and its application to the construction of the refined analytic torsion on odd dimensional compact manifolds with boundary. (Joint work with Yoonweon Lee)

Tee-How Loo  (University of Malaya, Malaysia) Title: Real Hypersurfaces in a Complex Space Form with h-parallel Shape Operator Abstract: Let M be a real hypersurface of a Riemannian manifold. The shape operator A of M is said to be parallel if it satisfies the condition ÑA = 0, where Ñ is the Levi-Civita connection on M. In particular, when the ambient space is a non-flat complex space form, it is well known that there exists no real hypersurfaces M with parallel shape operator. This leads to the concept of h -parallelism on the shape operator, that is, by restricting the parallelism condition of the shape operator to the holomorphic distribution. Many results concerning real hypersurfaces in a complex space form with h-parallel shape operator have been obtained in the past few decades. Nevertheless, real hypersurfaces in a complex space form with h-parallel shape operator have yet classified. In this paper, we shall first review some known results on real hypersurfaces in a non-flat complex space form space and then we give a partial characterization of real hypersurfaces in a non-flat complex space form space with h-parallel shape operator.

Makoto Narita  (Okinawa National College of Technology, Japan) Title: On spherically symmetric gravitational collapse in the Einstein-Gauss-Bonnet theory Abstract: We study global properties of spherically symmetric and asymptotically flat spacetimes with (nonlinear) scalar fields in the Einstein-Gauss-Bonnet (EGB) theory. The low energy limit of the heterotic string theory yields the quadratic curvature terms and a special combination of the terms is required to be the Gauss-Bonnet term to be ghost-free for graviton. Since the higher curvature terms would dominate in the region of strong gravitational fields, to study the nature of gravitational collapse and singularity in the EGB theory is meaningful to understand the stringy corrected theory of gravity. It is proven that the existence of a trapped surface implies the future completeness of future null infinity in the EGB theory. Also, we prove by showing a global existence theorem for the system of the EGB and scalar field equations in the domain of outer communications that the first singularities arising in the non-trapped region should necessarily be the centre.

Takashi Okayasu  ( Ibaraki University, Japan) Title: A construction of complete hypersurfaces with constant scalar curvature in the Euclidean space Abstract: We only have few examples of complete hypersurfaces with constant scalar curvature in the euclidean spaces. The first purpose of my talk is to construct a new family of complete hypersurfaces with constant positive scalar curvature in the euclidean spaces by the method of equivariant differential geometry. The second purpose is to show that we can perturb the half end of the complete rotational hypersurface with zero scalar curvature while keeping scalar curvature zero.

Mei-Lin Yau  (National Central University, Taiwan) Title: Monodromy groups of Lagrangian tori in the symplectic 4-space Abstract: In this talk we will describe the monodromy groups of Clifford tori and Chekanov tori associated to various types (smooth/Lagrangian/Hamiltonian) of self-isotopies. We will also present an algebraic proof that a monotone Clifford torus and a Chekanov torus are distinct up to Hamiltonian isotopies. This is done by comparing the Hamiltonian monodromy groups from the Maslov class perspective.

Takumi Yokota  (University of Tsukuba, Japan) Title: Perelman's reduced volume and gap theorem for the Ricci flow Abstract: In this talk, we consider the asymptotic limits of Perelman's reduced volume for ancient solutions to the Ricci flow. We will show that if the asymptotic reduced volume is sufficiently close to that of the Gaussian soliton, then the ancient solution must be isometric to the Euclidean space for all time. As an application, the case of the gradient shrinking Ricci solitons is discussed. Then we obtain a gap theorem for them, which proves the conjecture of Carrillo and Ni.