Fusion of Integrable Systems and Geometry

Date 17(Fri)14:00-19(Sun)16:00 April, 2009
Place Mathematical Institute, Tohoku University (東北大学理学部数学):Kawai Hall (川井ホール)

講演者(Speaker) Quo Shin Chi (Washington U in St Louis. Taiwan N. U), Josef Dorfmeister (Mueunchen TU), K. Hasegawa (長谷川浩司 Tohoku U), Hui Ma (Tsinghua U, OCAMI), J. Inoguchi (井ノ口順一 Yamagata U), Y. Kawakami (川上 裕 Kyushu U), Shimpei Kobayasi(小林真平 Hirosaki U), Y. Nohara (野原雄一 Tohoku U), T. Sakai (酒井高司 TMU), S. Udagawa (宇田川誠一 Nihon U), W. Rossman (Kobe U), Sumio Yamada (山田澄生 Tohoku U)

Quo-Shin Chi (Washington U in St Louis), I
"A new look at Condition A"
In this talk, I will first quickly review the recent progress on the classification of isoparametric hypersurfaces in spheres, which leaves open only the four exceptional cases of four principal curvatures with multiplicity pairs (4,5), (3,4), (7,8), (6,9). Ozeki and Takeuchi introduced Conditions A and B to construct inhomogeneous examples of isoparametric hypersurfaces with four principal curvatures of multiplicity pairs (3,4r) and (7,8r) in spheres. Later Dorfmeister and Neher showed that Condition A alone implies the isoparametric hypersurfaces is one of the examples constructed by Ferus, Karcher and Munzner that generalized those of Ozeki and Takeuchi. It appears Condition A may hold the key to the classification of the multiplicity pairs (3,4) and (7,8). Dorfmeister and Neher's method is algebraic in nature and their proof for the multiplicity pairs (3,4) and (7,8) relies on the classification result of McCrimmon about composition triples. We will present a fairly short and straightforward classification proof of the result of Dormeister and Neher based on more geometric considerations, in which the octonian algebra plays a decisive role.
Quo-Shin Chi (Washington U.in St. Louis), II
"On a problem of Kuiper"
Kuiper raised the question whether a taut submanifold in Euclidean space is algebraic, i.e., is an open subset of an irreducible component of a real algebraic variety in the ambient Euclidean space. I will show the answer is affirmative.
Josef Dorfmeister (Muenchen TU)
"Planar CMC 4-noids of constant mean curvature"
There are two parts to this talk: The first part discusses Fuchsian equations from the point of view of H.A.Schwarz. This means to consider a fundamental system u,v of solutions to a second order Fuchsian ODE and to discuss the polygon in the extended complex plane obtained as boundary curve to the image of the upper half-plane under the quotient map u/v. This polygon consists of circular arcs. We will characterize what it means that all monodromy matrices of the Fuchsian equations are unitary. In the second part we will apply ideas of Hilb and associate with the circular arcs of the Schwarz polygon planes intersecting the sphere in those circular arcs. Consequently, arcs meeting at some vertex create some axis via the the intersection of planes. We will characterize in terms of intersecting axes and the sign of some coefficient in some normalized Fuchsian equation what it means that all monodromy matrices are unitary. As a final application we will discuss what this means for the construction of planar CMC 4-noids.
長谷川浩司 Koji Hasegawa (Tohoku U.)
"Quantizing the discrete Painleve VI equation"
Painleve equations have rich symmetry known as "Baecklund transformations" which in fact generate affine Weyl groups. In the VIth case, the group is of type D_4^(1) and there is a symmetry-preserving time discretization by Jimbo and Sakai. We will quantize this, i.e. construct a non-commutative version of their discrete VIth equation. Two ways to quantize: the symmetry point of view and the monodromy-preserving point of view. It turns out that these two provides the same equation.
Hui Ma (Tsinghua U. OCAMI)
"Hamiltonian stability of the Gauss images of homogeneous isoparametric"
The volume minimizing problem of Lagrangian submanifolds in Kaehler manifolds under Hamiltonian deformations arises as a constraint variational problem in the intersection of Riemannian Geometry and Symplectic Geometry. As a generalization of minimal submanifolds and usual stability, it defines Hamiltonian minimal submanifolds and Hamiltonian stability. It is interesting to study the construction of Hamiltonian minimal submanifolds in specific Kaehler manifolds and their Hamiltonian stability. In this talk, we focus on minimal Lagrangian submanifolds in complex hyperquadrics, obtained from Gauss images of isoparametric hypersurfaces in unit spheres. We shall discuss the Hamiltonian stability in the homogenous case. This talk is based on the joint work with Prof. Yoshihiro Ohnita.
井ノ口順一 Jun-ichi Inoguchi (Yamagata U.)
"Constant mean curvature surfaces in hyperbolic 3-space"
I will discuss a generalised Weierstrass representation for constant mean curvature surfaces in hyperbolic 3-space with mean curvature less than 1. This is a joint work with Josef Dorfmeister and Shimpei Kobayashi.
川上 裕 Yu Kawakami (Kyushu U.)
"On the upper bound of the number of exceptional values of the hyperbolic Gauss map"
In this talk, we will give the upper bound of the number of exceptional values of the hyperbolic Gauss map of algebraic CMC-1 surfaces and flat fronts with regular ends in the hyperbolic three-space and discuss the sharpness of the estimates.
小林真平 Shimpei Kobayashi (Hirosaki U.)
"Harmonic trinoids in complex projective spaces"
I shall give a talk about a construction of harmonic maps from a thrice punctured Riemann sphere into complex projective spaces using loop group techniques and theory of ordinary differential equations.
野原雄一 Yuichi Nohara (Tohoku U.)
"Toric degenerations of Gelfand-Cetlin systems and potential functions"
The Gelfand-Cetlin system is a completely integrable system on a flag manifold of type A, whose moment polytope is called the Gelfand-Cetlin polytope. It is also known that the flag manifold has a degeneration into a toric variety corresponding to the Gelfand-Cetlin polytope. We show that the Gelfand-Cetlin system can be deformed into a moment map of a torus action on the toric variety. We also discuss an application to mirror symmetry. This is based on a joint work with T. Nishinou and K. Ueda.
山田澄生 Sumio Yamada (Tohoku U.)
"Weil-Petersson geometry of Teichmuller-Coxeter complex"
The Weil-Petersson metric on a Teichmuller space is well known to be incomplete. We construct a geodesic completion of the Teichmuller space, with help of the Coxeter complex formalism, where each copy of the Teichmuller space acts as a simplex. We deduce a finite rank property from such a construction.
Wayne Rossman (Kobe U.)
"Discrete flat and linear Weingarten surfaces in hyperbolic 3-space"
This is a joint work with Tim Hoffmann (Munich Technical University), Takeshi Sasaki and Masaaki Yoshida. Previous works have led to definitions of discrete minimal surfaces in Euclidean 3-space (Bobenko and Pinkall) and discrete CMC 1 surfaces in hyperbolic 3-space (Hertrich-Jeromin). Taking inspiration from those works, and from works by Galves-Martinez-Milan and Kokubu-Umehara-Yamada which give representations for smooth flat and smooth linear Weingarten surfaces in hyperbolic 3-space, we find discretizations for those surfaces. We will present some resulting theorems that justify our choice of definitions, and we will examine notions of singularities and caustics on discrete flat surfaces.
酒井高司 Takashi Sakai (Tohoku Metrop. U.)
"Tight Lagrangian surfaces in S^2×S^2"
In 1991, Y.-G. Oh introduced the notion of tightness of closed Lagrangian submanifolds in Hermitian symmetric spaces. It is known that any real forms in a compact Hermitian symmetric space are tight. In this talk, we determine all tight Lagrangian surfaces in S^2×S^2. Latter, we shall discuss global tightness of real forms in Hermitian symmetric spaces. In particular, we will show that globally tight Lagrangian surfaces in S^2×S^2 are nothing but real forms. This talk is based on a joint work with Hiroshi Iriyeh.
宇田川 誠一 Seiichi Udagawa (Nihon U.)
"Affine spheres of finite type and Symes method"
We construct an extended framing for Blaschke immersion of affine spheres in A^3 and may give all Blaschke immersions of finite type by Symes method. Moreover, DPW-method also gives a Weierstrass-Kenmotsu type representation for Blaschke immersions of affine spheres. This is a joint work with J. Inoguchi.

R. Miyaoka(宮岡礼子) Tohoku U.

Contact :
r-miyaok (at) math.tohoku.ac.jp

This Workshop is supported by Grants-in-Aid for Scientific Research 19204006 (R. Miyaoka).
製作 宮岡 Last updated on 24 March 2009