# Differential Geometry Seminar (2009)

As a project of OCAMI, we shall promote the seminar on differential geometry in the wide sense of including the areas related to geometric analysis, topology, algebraic geometry, mathematical physics, integrable systems, information sciences etc.

Contact | Yoshihiro Ohnita Shin Kato Department of Mathematics Osaka City University Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, JAPAN |
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TEL | 06-6605-2617 (Ohnita) 06-6605-2616 (Kato) |

ohnita@sci.osaka-cu.ac.jp shinkato@sci.osaka-cu.ac.jp |

Speaker | Kurosu SANAE (Dept. of Math., Tokyo University of Science) |
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Title | Pluriharmonic maps and tt*-structures |

Date | March 31 (Wed.) 2010, 13:30～15:00 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | This is based on a collaboration work with Dr. Katsuhiro Moriya in Tsukuba University. tt*-geometry has a deep relation with "integrable system" "variations of Hodge structure" "pluriharmonic maps" and is studied by C.Hertling, V.Cortes, L.Schafer, for example. Especially, a correspondence between (metric) tt*-bundles and pluriharmonic maps into some Riemannian symmetric space is studied by L. Schafer. For a constant mean curvature(CMC) surface, we can construct a tt*structure by using the theory of quaternionic vector bundle. We can also construct a metric tt*-bundle structure from a harmonic map into 2-sphere since it is related to the Gauss map of a constant mean curvature surface. In this talk, we will introduce the construction of a tt*-bundle from a pluriharmonic maps to an n-sphere in Clifford algebra as a generalization of the above result. |

Speaker | Daisuke NAKAJYO (Kyushu University) |
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Title | The representation formula for indefinite improper affine spheres |

Date | March 31 (Wed.) 2010, 10:40～12:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | Improper affine spheres are important objects in affine differential geometry. The study of representation formulae for them has long history, and it has been still developed now on. In this talk, I will construct a new representation formula for indefinite improper affine spheres in terms of para-holomorphic functions and explain some related topics. |

Speaker | Shinobu Fujii（Hiroshima University D3） |
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Title | On isoparametric hypersurfaces in spheres and moment maps |

Date | March 18 (Tｈu.) 2010, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | An isoparametric function on spheres is obtained by the restriction of a Cartan-Munzner polynomial on the spheres. In this talk, we consider the isotropy representations of irreducible Hermitian symmetric spaces of rank two, and we explain that (weighted) square norms of their moment maps are Cartan-Munzner polynomials, which define homogeneous isoparametric hypersurfaces in spheres with four distinct principal curvatures. Our expectation is that every isoparametrc hypersurface in spheres with four distinct principal curvatures is related to a moment map for a certain group action. |

Speaker | Kazuhiro Shibuya（Hiroshima University） |
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Title | The singularity of differential systems |

Date | March 2 (Tue.) 2010, 16:30～18:00 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | Japanese page only |

Speaker | Takahiro Noda （Nagoya University） |
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Title | A geometric study of type-changing equations via differential systems |

Date | March 2 (Tue.) 2010, 10:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | Type-changing equations are belong to single second order PDEs,and these are very interesting objects. In this talk, I explain our results which we obtained until now for a study of these equations. This talk is based on a joint work with Kazuhiro Shibuya (Hiroshima University). |

Speaker | Prof. Sigmundur.Gudmundsson（Lund University, SWEDEN） |
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Title | Harmonic morphisms from Lie groups and symmetric spaces (2nd lecture) |

Date | Feb. 18 (Thu.) 2010, 16:20～17:50 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | We will discuss the existence problem for harmonic morphisms between Riemannian manifolds. In particular we shall focus our attention on recent results on complex-valued harmonic morphisms from Lie groups and symmetric space. |

Speaker | Prof. Sigmundur.Gudmundsson（Lund University, SWEDEN） |
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Title | Harmonic morphisms, complex analysis and how to generalize it (1st lecture) |

Date | Feb. 18 (Thu.) 2010, 10:40～12:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | It is a well-known result in classical complex analysis that every holomorphic or anti-holomorphic function is harmonic and conformal. It is easy to see that they also pull back harmonic real-valued functions to harmonic functions. This last property actually characterizes the holomorphic and anti-holomorphic functions in the complex plane. Harmonic morphisms are maps (M,g) -> (N,h) between Riemannian manifolds which pull back harmonic functions to harmonic functions. They have many properties similar to those of holomorphic functions which they generalize. We will give a general introduction to the theory of harmonic morphisms |

Speaker | Yu KAWAKAMI ( Dept. of Math., Kyushu Univ. ) |
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Title | Geometry of fronts - from the viewpoint of value distribution of Gauss map - (Part 2) |

Date | Feb. 11 (Thu.) 2010, 10:40～12:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | In this seminar, we give a ramification estimate for another Gauss map of weakly complete flat fronts in hyperbolic three-space. As an application, we provide a new proof of the characterization of (weakly) complete flat surfaces in hyperbolic three-space. Moreover, we discuss a similar argument for improper affine spheres in three-dimensional affine space. This is joint work with Daisuke Nakajo (Kyushu university). |

Speaker | Yu KAWAKAMI ( Dept. of Math., Kyushu Univ. ) |
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Title | Geometry of fronts - from the viewpoint of value distribution of Gauss map - (Part 1) |

Date | Feb. 11 (Thu.) 2010, 10:40～12:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | In this seminar, we will talk about the hyperbolic Gauss maps of complete flat fronts in hyperbolic three-space. In particular, we give the upper bound for the number of exceptional values of them in some topological cases. |

Speaker | Hirotake KURIHARA ( Dept. of Math., Tohoku Univ. ) |
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Title | On designs in symmetric spaces |

Date | Feb. 10 (Wed.) 2010, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | The theory of designs is originally started from a combinatorial study of some collections of subsets of finite sets which are called block designs. Delsarte, Goethals and Seidel (1977) gived the definition of designs in spheres in Euclidean spaces and proved that theory of spherical designs is similar to theory of block designs. Moreover the present design theory is expended into compact symmetric spaces. Here, a finite subset of a compact symmetric space is called a design if the averaging over the subset is equals to the averaging over the compact symmetric space for any function on the compact symmetric space in a specific class. The concept of designs appears in our life, for instance in meteorology, and the theory is also related to several mathematical areas. In this talk, I will introduce topics of the existent of designs and the lower bounds for the size of designs while giving some examples of designs. |

Speaker | Professor Ian M. Anderson（Utah State University） |
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Title | Group invariant solutions and symmetric criticality |

Date | Dec. 15 (Tue.) 2009, 16：30～18：00 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | Japanese page only |

Speaker | Yukinori YASUI (Department of Physics, Osaka City University) |
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Title | Geometry of higher dimensional black holes and compact Einstein manifolds |

Date | Nov. 18 (Wed.) 2009, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | In this talk we show that inhomogeneous compact Einstein manifolds are explicitly constructed from higher dimensional black hole solutions. This is a generalization of Page's construction. He found the first inhomogeneous Einstein metric on the connected sum CP2 # CP2 of two copies of CP2 with opposite orientation. Here we classify higher dimensional black hole solutions admitting conformal Killing-Yano tensor. Using these solutions we construct inhomogeneous Einstein manifolds. This method offers a unified viewpoint for a past study and gives new Einstein metrics. |

Speaker | Tomonori NODA ( Osaka Dental University ) |
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Title | Dually flat spaces, canonical structures and moment maps |

Date | Nov. 11 (Wed.) 2009, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | The image of the moment map for the torus action on the two dimensional complex projective space is a triangle in a plane. We can identify this triangle with the all probability distributions on the set which consists of three points. In this situation, we have a duality between the Fubini-Study metric on the projective space and the Fisher metric on the moment image. We consider this duality from Information geometric viewpoint. We show that this duality holds for any dually flat space. |

Speaker | Nobutaka BOUMUKI (Osaka City University Advanced Mathematical Institute) |
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Title | Reflective submanifolds of affine symmetric spaces II |

Date | Oct. 21 (Wed.) 2009, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | This talk is a sequel to my talk in the mathematical conference held at Osaka City University on 20 September 2006. I plan to talk about how study of reflective submanifolds has progressed. Besides, I plan to explain that the Noda correspondence is a correspondence among pseudo-hermitian symmetric spaces, Lagrangian reflective submanifolds, para-hermitian symmetric spaces and paracomplex reflective submanifolds. |

Speaker | Yoshihiro OHNITA ( Osaka City University ) |
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Title | On Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces, II |

Date | Oct. 14 (Wed.) 2009, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | This talk is the Part II of my talk at OCAMI Differential Geometry Seminar, July 16, 2008. In this talk I shall report a progress in my joint work with Dr. Hui Ma (Tsinghua University, Beijing) on Lagrangian submanifolds in complex hyperquadrics, which is a compact Hermitian symmetric spaces of rank 2. For any compact isoparametric hypersurface in the unit standard sphere, the image of its Gauss map is a compact minimal Lagrangian submanifold embedded in a complex hyperquadric. We have determined the Hamilitonian stability of ALL compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as the Gauss images of homogeneous isoparametric hypersurfaces in spheres. |

Speaker | Fuminori NAKATA (Graduate School of Science and Engineering, Tokyo Institute of Technology) |
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Title | Wave equations and the LeBrun-Mason correspondence |

Date | Oct. 7 (Wed.) 2009, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | The twistor theory concerning holomorphic disks, developed by LeBrun and Mason, is now progressing steadily. There are several types of LeBrun-Mason theory, and one of them concern self-dual Zollfrei conformal structures. Here, an indefinite metric (or an indefinite conformal structure) on a manifold is called Zollfrei if and only if all the null geodesics are closed. On the other hand, Tod and Kamada independently constructed explicit examples of indefinite self-dual metrics. In this talk, we show that the Tod-Kamada metrics are all Zollfrei and that the corresponding LeBrun-Mason twistor space is explicitly written down. This correspondence is given by using a generalization of Radon transform, and in the way of the proof we obtain explicit descriptions of solutions of the wave equation and the monopole equation on the de Sitter 3-space via integral transform. (preprint: arXiv:0907.0928) |

Speaker | Miyuki KOISO ( Nara Women's University & PRESTO, JST ) |
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Title | Uniqueness for "small" CMC (constant mean curvature) surfaces with boundary |

Date | July 29 (Wed.) 2009, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | Let C be a Jordan curve in the closed ball with radius R in the three-dimensional Euclidean space. Let H be a nonzero constant of which the absolute value is smaller than 1/R. Then, there exist at least two surfaces with constant mean curvature H ( CMC-H ) which are spanned by C, and one of them is contained in the closed ball mentioned above. In this talk, we give a sharp sufficient condition for C so that the "small" CMC-H surface spanned by C is unique. As a corollary, we obtain the following theorem by Rado: If C has a one-to-one orthogonal projection to a convex planar curve, then the minimal surface spanned by C is unique. |

Speaker | Nobutaka BOUMUKI ( OCAMI ) |
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Title | On relation between pseudo-Hermitian symmetric spaces and para-Hermitian symmetric spaces |

Date | Juｌｙ 5 (Wed.) 2009, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | Assertion in this talk is based on joint work with Dr. Tomonori Noda. Our main purpose in this talk is to assert that (I) any simple pseudo-Hermitian symmetric space G/R and Lagrangian reflective submanifold of G/R bring about a simple para-Hermitian symmetric space, and (II) any simple para-Hermitian symmetric space is brought about by a simple pseudo-Hermitian symmetric space G'/R' and a Lagrangian reflective submanifold of G'/R'. That is a correspondence between Kaehler or pseudo-Kaehler manifolds and para-Kaehler manifolds in the class of simple affine symmetric space. |

Speaker | Takahiko YOSHIDA ( Dept. of Math., Meiji Univ. ) |
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Title | Acyclic polarizations and localization of Riemann-Roch numbers |

Date | June 10 (Wed.) 2009, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | Joint work with Mikio Furuta (The University of Tokyo) and Hajime Fujita (Gakushuin University) For a prequantizable closed symplectic manifold a Riemann-Roch number is defined to be the index of a spin^c Dirac operator with coefficients in the prequantization line bundle. Suppose the prequantizable symplectic manifold is equipped with a structure of a Lagrangian fiber bundle. A fiber of the Lagrangian fiber bundle is said to be Bohr-Sommerfeld if the restriction of the prequantization line bundle to the fiber is trivially flat. Bohr-Sommerfeld fibers appear discretely. Then Andersen showed that the Riemann-Roch number is equal to the number of Bohr-Sommerfeld fibers. In the case of Lagrangian fiber bundles with singular fibers this equality is also observed for several examples, such as, moment maps of toric varieties, the Gelfand-Cetlin system for unitary groups, Goldman's Hamiltonian system on the moduli space of flat SU(2)-bundle on a Riemann surface, by computing and comparing both sides. In this talk, for this phenomenon we show that if a prequantizable symplectic manifold admits a structure of a singular Lagrangian fibration, the Riemann-Roch number is localized on nonsingular Bohr-Sommerfeld fibers and singular fibers. The technique used here is the localization of Witten-type. Reference: arXiv:0804.3258 |

Speaker | Pablo Mira Carrillo (Universidad Politecnica de Cartagena, SPAIN) |
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Title | CMC surfaces in homogeneous 3-manifolds |

Date | Apr. 16 (Thu.) 2009, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | It is a natural question to consider CMC surfaces in other ambient spaces than Euclidean space. Thanks to the holomorphicity of the Abresch-Rosenberg differential (closely related to the Hopf differential), the study of CMC surfaces in the simply-connected Riemannian homogeneous 3-manifolds (or, rather, the eight Thurston geometries) becomes accessible. This is the topic of this talk. |