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{(Japanese)

# Differential Geometry and Geometric Analysis

 Date: March 9 (Mon)- March 10 (Tue), 2009 Place: Lecture Room 3040, Department of Mathematics, Graduate School of Science, Osaka City University

 Organizers: Yng-Ing Lee (NTU & TIMS), Chang-Shou Lin (NTU, TIMS Director), Shin Kato (OCU), Futoshi Takahashi (OCU), Martin Guest (TMU & visiting prof. of OCU), Akio Kawauchi (OCU, OCAMI Director) Yoshihiro OhnitaiOCU, chairpersonj

 Sponsors F Osaka City University Advanced Mathematical Institute (OCAMI). National Taiwan University (NTU), Taida Institute for Mathematical Sciences

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 Invited researchers F Professor Yng-Ing Lee iNational Taiwan University, Taiwan) Professor Chang-Shou LiniNational Taiwan University, Taiwan) Professor Shu-Cheng ChangiNational Taiwan University, Taiwan) Professor Derchyi WuiAcademia Scinica, Taiwan) Professor River ChiangiNational Cheng Kung University, Taiwan) Professor Quo-Shin Chi (Washington University, USA & National Taiwan University, Taiwan) Professor Jost Hinrich Eschenburg (University of Augsburg, Germany) Professor Hajime OnoiScience University of Tokyo, Japan) Professor Manabu AkahoiTokyo Metropolitan University, Japan)

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 Program (speakers, titles and abstracts)F program , abstract March 9 (Mon) 9:50-10:00@Opening Speech by OCAMI Director Akio Kawauchi 10:00-11:00@Yoshihiro Ohnita (Osaka City University, Japan) Differetial geometry of Lagrangian submanifolds and Hamiltonian variational problems. Abstract: In this talk I will explain Hamiltonian minimality and Hamiltonian stability problem for Lagrangian submanifolds in specific Kaehler manifolds and I will mention recent results in my joint works with Hui Ma on minimal Lagrangian submanifolds in complex hyperquadrics obtained as the Gauss images of isoparametric hypersurfaces in the standard unit sphere. 11:10-12:10@Yng-Ing LeeiNational Taiwan University, Taiwan) "On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds" In this talk, I will report my recent joint work with D. Joyce and R. Schoen. Let $(M,\omega)$ be a compact symplectic $2n$-manifold, and g be a Riemannian metric on $M$ compatible with $\omega$. For instance, g could be Kaehler, with Kaehler form $\omega$. Consider compact Lagrangian submanifolds $L$ of $M$. We call $L$ Hamiltonian stationary, or H-minimal, if it is a critical point of the volume functional under Hamiltonian deformations. Our main result is that if $L$ is a compact, Hamiltonian stationary Lagrangian in C^n which is Hamiltonian rigid, then for any $(M,\omega,g)$ as above there exist compact Hamiltonian stationary Lagrangians $L'$ in M contained in a small ball about some point p and locally modelled on tL for small t>0, identifying $M$ near p with $C^n$ near 0. If L is Hamiltonian stable, we can take $L'$ to be Hamiltonian stable. Applying this to known examples in $C^n$ shows that there exist families of Hamiltonian stable, Hamiltonian stationary Lagrangians diffeomorphic to $T^n$, and to $S^1 x S^{n-1}/ Z_2$, and with other topologies, in every compact symplectic 2n-manifold $(M,\omega)$ with compatible metric $g$. 13:30-14:30@River Chiang (National Cheng Kung University, Taiwan) "On the construction of certain 6-dimensional Hamiltonian SO(3) manifolds" Abstract: In this talk, I will discuss a construction of 6-dimensional Hamiltonian SO(3) manifolds. In 2005, I gave the invariants to distinguish these manifolds up to equivariant symplectomorphisms, which constitute the uniqueness part of the classification. The construction in this talk is one step toward the existence part. 14:50-15:50@Futoshi Takahashi (Osaka City University) "On an eigenvalue problem related to the critical Sobolev exponent: variable coefficient case" Abstract: PDF 16:00-17:00 Chang-Shou Lin@iNational Taiwan University, Taiwan) "Green function and Mean field equations on torus" Abstract: The Liouville equation is an integrable system. Locally, solutions can be written explicitly by Liouville theorem. So, it is interesting to study solution structure globally. In my talk, I will show you how applying Elliptic function theory and PDE technique together to study the equation. As application of our theory, we prove the Green function of torus has at most five critical points. March 10 (Tue) 10:00-11:00 Jost Hinrich Eschenburg@ (University of Augsburg, Germany) "Constant mean curvature surfaces and monodromy of Fuchsian equations" Abstract: We will discuss classical theory (going back to H.A. Schwarz) of certain Fuchsian equations, i.e. second order linear ODEs $$y'' + py' + qy = 0$$ where $p,q$ are real rational functions with only regular singularities lying on the real line. In particular we investigate in which cases the monodromy group is (up to conjugation) contained in the isometry group of either the 2-sphere or euclidean or hyperbolic plane. As an application we study punctured spheres of constant mean curvature in euclidean 3-space where all punctures lie on a common circle. 11:10-12:10 Derchyi Wu (Academia Scinica, Taiwan) "The Cauchy Problem of the Ward Equation" Abstract: PDF 13:30-14:30@Manabu AkahoiTokyo Metropolitan University, Japan) "Lagrangian mean curvature flow and symplectic area" Abstract: I'll explain a very easy observation of Lagrangian mean curvature flow in an Einstein-Kaehler manifold and the symplectic area of smooth maps from a Riemann surface with boundary on the flow. @@@ 14:50-15:50@Hajime OnoiScience University of Tokyo, Japan) "Variation of Reeb vector fields and its applications" Abstract: In this talk, we give some applications of the variation of Reeb vector fields of Sasaki manifolds: Given a Fano manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is the Futaki invariant which is an obstruction for the existence of K\"ahler-Einstein metrics. We show that these obstructions are obtained as derivatives of the Hilbert series. Especially, in toric case, we can compute the Hilbert series and its derivative using the combinatorial data of the image of the moment map. This observation should be regarded as an extension of the volume minimization of Martelli, Sparks and Yau.@@@@ @@@@ 16:00-17:00 Shu-Cheng ChangiNational Taiwan University, Taiwan) "The CR Bochner formulae and its applications" Abstract: (i) In first half, we will prove the CR analogue of Obata's theorem on a closed pseudohermitian manifold with vanishing pseudohermitian torsion. The key step is a discovery of CR analogue of Bochner formula which involving the CR Paneitz operator and nonnegativity of CR Paneitz operator. This is a joint work with H.-L. Chiu which to appear in Math. Ann. and JGA. (ii) In second half, we obtain a new Li-Yau-Hamilton inequality for the CR Yamabe flow. It follows that the CR Yamabe flow exists for all time and converges smoothly on a spherical closed CR 3-manifold with positive Yamabe constant and vanishing torsion. This is a joint work with H.-L. Chiu and C.-T. Wu which to appear in Transactions of AMS.

 Concentrated Lectures by Professor Jost Hinrich Eschenburg on Harmonic Map Theory Dates : March 12 (Thu)- March 13 (Fri) am 10:00-12:00 (two lectures) Place : Lecture Room 3040, Department of Mathematics, in the buliding of Faculty of Science. Title : Pluriharmonic Maps and Submanifolds Abstract : PDF Lecture Notes: PDF

 Concentrated Lectures by Professor Quo-Shin Chi on Isoparametric Submanifold Theory Dates : March 12 (Thu)- March 13 (Fri) pm 15:00-17:00 (two lectures) Place : Lecture Room 3040, Department of Mathematics, in the buliding of Faculty of Science. Title : The Isoparametric Story Abstract : I will talk about the almost complete classification of isoparametric hypersurfaces in spheres, and, if time allows, end with a new look at two of the four unclassified cases.

Suggestion to SpeakersF At the lecture room there are enough blackboards, the computer projector and the visualizer. Please prepare your talk using them.

 LinkF Osaka City University Advanced Mathematical Institute (OCAM I) Department of Mathematics, Osaka City Univercity National Taiwan University Taida Institute for Mathematical Sciences (TIMS) TIMS One-Day Workshop on Differential Geometry (April 1, 2008) Poster

 Support : JSPS Grant-in-Aid for Scientific Research (A) No.17204006 "Research via the Infinite Dimensional Methods in Submanifold Theory" (Head researcher : Yoshihiro Ohnita)

Contact : ie-mailj
Yoshihiro OhnitaF ohnita (at) sci.osaka-cu.ac.jp
Shin Kato F shinkato (at) sci.osaka-cu.ac.jp
Futoshi Takahashi F futoshi (at) sci.osaka-cu.ac.jp
Osaka City University Advanced Mathematical Institute & Department of Mathematics, Osaka City University
3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585 JAPAN

Constructed by T. Noda (ODU) Last updated on 1/August/2009