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Program (speakers, titles and abstracts)：
program ,
abstract
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March 9 (Mon) |
| 9:50-10:00 Opening Speech by OCAMI Director Akio Kawauchi |
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| 10:00-11:00 Yoshihiro Ohnita (Osaka City University, Japan) |
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Differetial geometry of Lagrangian submanifolds
and Hamiltonian variational problems.
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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.
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| 11:10-12:10 Yng-Ing Lee（National Taiwan University, Taiwan) |
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"On the existence of Hamiltonian stationary Lagrangian submanifolds
in symplectic manifolds" |
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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 $g$.
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| 13:30-14:30 River Chiang (National Cheng Kung University, Taiwan) |
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"On the construction of certain 6-dimensional Hamiltonian SO(3) manifolds"
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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.
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| 14:50-15:50 Futoshi Takahashi (Osaka City University) |
| "On an eigenvalue problem related to the critical Sobolev
exponent: variable coefficient case" |
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Abstract:
PDF
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| 16:00-17:00 Chang-Shou Lin （National Taiwan University, Taiwan) |
| "Green function and Mean field equations on torus" |
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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.
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| March 10 (Tue) |
| 10:00-11:00 Jost Hinrich Eschenburg
(University of Augsburg, Germany) |
| "Constant mean curvature surfaces and monodromy
of Fuchsian equations" |
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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.
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| 11:10-12:10 Derchyi Wu (Academia Scinica, Taiwan) |
| "The Cauchy Problem of the Ward Equation" |
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Abstract:
PDF
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| 13:30-14:30 Manabu Akaho（Tokyo Metropolitan University, Japan) |
| "Lagrangian mean curvature flow and symplectic area" |
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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.
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| 14:50-15:50 Hajime Ono（Science University of Tokyo, Japan) |
| "Variation of Reeb vector fields and its applications" |
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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.
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| 16:00-17:00 Shu-Cheng Chang（National Taiwan University, Taiwan) |
| "The CR Bochner formulae and its applications" |
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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.
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