# Differential Geometry Seminar (2008)

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 | Kotaro Kawai (University of Tokyo, M2) |
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Title | Torus invariant Special Lagrangian Submanifolds in the Canonical Bundle of Toric Fano Manifolds |

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

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

Abstract | We construct torus invariant special Lagrangian submanifolds in the canonical bundle of toric Fano manifolds. Special Lagrangian submanifolds are certain classes of minimal submanifolds and defined in Calabi-Yau manifolds. Using the Ricci-flat metric constructed by Futaki, we give the Calabi-Yau structure to the canonical bundle. Then, using the moment map technique developed by Ionel and Min-Oo, we construct special Lagrangian submanifolds. |

Speaker | Jost Hetrich Eschenburg（University of Augsburg, Germany) |
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Title | Penrose Tiling and its Generalization |

Date | March 18 (Wed.) 2009, 13:00～14:00 |

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

Abstract | I shall offer an extra talk for young students, not on differential geometry but still on geometry: I have studied quite intensively the Penrose tiling, a periodic pattern obeying very strong rules. I shall tell this in a completely elementary way or in a more advanced way (or a mixture of both). I shall make a few observations which are not so common. I think that such regular patterns could be of some interest in the land of origami (though there is no direct relationship). |

Speaker | Yu Kawakami (Dept. of Math., Kyushu university & OCAMI) |
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Title | Non-existence of minimal surfaces with embedded planar ends |

Date | March 12 (Thu.) 2009, 17:30～19:00 |

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

Abstract | We will talk about the algebraic characterization of embedded planarends of minimal surfaces and its applications by Kusner and Schmitt. |

Speaker | Yukinori Yasui (Dept. of Physics, Osaka City University) |
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Title | Einstein Manifold with Conformal Killing-Yano tensor |

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

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

Abstract | We talk about Einstein manifolds admitting a conformal Killing-Yano(CKY) tensor. The black hole spacetime is taken into consideration for a concrete example. We give a local classification of the spacetimes with a closed rank-2 CKY tensor. By compactifying the spacetimes Einstein metrics are constructed on the sphere bundles over Kaehler manifolds. |

Speaker | Professor Toyoko Kashiwada |
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Title | Conformal Killing Tensors and Tensor Analysis |

Date | March 11 (Wed.) 2009, 10:40～12:10 |

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

Abstract | Japanese page only |

Speaker | Sanae KUROSU (Tokyo Metropolitan University) |
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Title | A charactrization of a special Kaehler manifold as an affine hypersphere and its representation formula |

Date | February 6 (Fri.) 2009, 10:40～12:10 |

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

Abstract | I shall provide an explanation of the following two papers: <\br> Realisation of special Kaehler manifolds as parabolic affine spheres O.Baues, V. Cortes Proc AMS 129 2403-2407<\br> A holomorphic representation formula for parabolic affine hyperspheres <\br> V. Cortes Special Kaehler manifolds can be characterized by improper affine hyperspheres. I shall explain about its affine differential geometric proof and the representation formula in terms of holomorphic functions given from the manifold structure. |

Speaker | Mark Hamilton ( Tokyo Univ. ) |
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Title | Geometric quantization of integrable systems |

Date | January 29 (Thu.) 2009, 16:30～18:00 |

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

Abstract | The theory of geometric quantization is one way of producing a "quantum system" from a "classical system," and has been studied a great deal over the past several decades. It also has surprising ties to representation theory. However, despite this, there still does not exist a satisfactory theory of quantization for systems with singularities. Geometric quantization requires the choice of a polarization; when using a real polarization to quantize a regular enough manifold, a result of Sniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld fibres. However, there are many types of systems to which this result does not apply. One such type is the class of completely integrable systems, which are examples coming from mechanics that have many nice properties, but which are nonetheless too singular for Sniatycki's theorem to apply. In this talk we will explore one approach to the quantization of integrable systems, and show a Sniatycki-type relationship to Bohr-Sommerfeld fibres. However, some surprising features appear, including infinite-dimensional contributions and strong dependence on the polarization. I will give at least a brief explanation of both geometric quantization and integrable systems, and hope to make the talk accessible to a general differential geometric audience. This is joint work with Eva Miranda. |

Speaker | Haizhong Li ( Tsinghua University, Beijing, China ) |
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Title | Willmore submanifolds in a Riemannian manifold |

Date | January 28 (Wed.) 2009, 14:40～16:10 |

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

Abstract | In this talk, we give a survey of geometry of Willmore submanifolds, which includes Willmore functional, Willmore conjecture, Willmore surfaces, Willmore hypersurfaces and Willmore submanifolds. |

Speaker | Yu Kawakami (Dept. of Math., Kyushu university & OCAMI) |
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Title | Bloch's principle and its application to theory of the Gauss map of minimal surfaces |

Date | January 23 (Fri.) 2009, 14:20～15:20 |

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

Abstract | Bloch's principle says that a family of meromorphic functions is likely to be normal if there is no non-constant entire functions with this property. By applying this, Antonio Ros gave a simple proof of the Fujimoto theorem on the best possible upper bound of the number of exceptional values of the Gauss map for complete minimal surfaces in Euclidean three-space. In this talk, I will explain these results. |

Speaker | Sanae KUROSU (Tokyo Metropolitan University) |
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Title | A cylinder theorem for an affine immersion from a (para-)complex manifold |

Date | January 23 (Fri.) 2009, 13:15～14:15 |

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

Abstract | It is known that any minimal (puriharmonic or (1,1)-geodesic) isometric hypersurface from a complete K\"ahler manifold to a Euclidean space is a cylindr (K. Abe "On a class of hypersurface of {R}^{n+1}" (Duke Math. J, \textbf{41} (1974), 865-874)). we study an affine immersion from a (para-)complex manifold with a certain condition for its affine fundamental form, which includes a (1,1)-geodesic affine immersion from a complex manifold. We also derive a cylinder theorem for such a hypersurface. |

Speaker | Yumiko KITAGAWA ( OCAMI ) |
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Title | Geodesics on sub-Riemannian manifolds |

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

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

Abstract | A sub-Riemannian manifold is a manifold endowed with a tangent distribution and a fibre inner product on the distribution. A distribution here means a linear subbundle of the tangent bundle. The Riemannian fibre metric is called a Carnot Caratheodry metric, which is related to the Optimal Control Theory. We suppose that a distribution is bracket generating (i.e., the collection of all vector fields generated by Lie brackets spans the whole tangent bunndle). In this talk we study geodesics on sub-Riemannian manifolds. |

Speaker | Shin NAYATANI ( Dept. of Math., Nagoya Univ. ) |
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Title | Fixed-point property of discrete groups and harmonic maps |

Date | January 14 (Wed.) 2009, 16:30～18:00 |

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

Abstract | A discrete group is a group consisting of countably infinite number of elements. When such a group acts isometrically on a metric space, it may happen that every element fixes a single point of the space; this phenomenon is the theme of this talk. Though looks quite special, such phenomenon was classically observed in Kazhdan's property (T) and in Margulis' superrigidity theorem. In this talk, I will give criterions for a discrete group to have fixed-point property for a large class of metric spaces (precisely, CAT(0) spaces, which are negatively curved in some sense). I also explain that a certain random group satisfies this criterion. The ingredient we use is a discrete analogue of the harmonic map which is well-known in differential geometry. I hope to explain how ideas in diffrential geometry and geometric analysis can be applied to the study of discrete groups as much as time permits. |

Speaker | Yuki IGUCHI ( Dept. of Math., Kanazawa Univ. ) |
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Title | Asymptotic behavior of Teichmuller geodesics near Thurston boundary |

Date | November 19 (Wed.) 2008, 14:40～16:10 |

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

Abstract | Teichmuller space is a set of all equivalence classes of conformal structures on a Riemann surface. This set is known to consist of all equivalence classes of hyperbolic metrics on the surface. In this talk, we introduce the distance function called Teichmuller distance on the set and the compactification by Thurston. Geodesics with respect to the distance are characterized by non-trivial holomorphic quadratic differentials. We examine asymptotic behavior of the geodesics near Thurston boundary. In particular, we observe Lenzhen's result which says that there are geodesics which do not have a limit in the boundary. |

Speaker | Miguel Ortega Titos (University of Granada, SPAIN) |
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Title | Marginally trapped surfaces in Minkowski 4-space which are invariant by some isometry groups |

Date | September 22 (Mon.) 16:00～17:30 |

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

Abstract | Basically, marginally trapped surfaces in a Lorentzian manifold are those spacelike surfaces whose mean curvature vector is lightlike. These surfaces are interesting in Physics, since the boundary of a region of the Universe containing black holes are foliated by such surfaces, among many other properties. We pay attention to those marginally trapped surfaces which are invariant by some 1-dimensional isometry subgroups of O(4,1). Depending on the chosen group, we describe the surfaces whose mean curvature vector is either lightlike or zero, and we study classical geometric properties on them. |

Speaker | Magdalena Caballero (University of Granada, SPAIN) |
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Title | A complete classification of rotational surfaces in L3 (the Lorentz-Minkowski 3-space) -Rotational Willmore surfaces in L3- |

Date | September 22 (Mon.) 14:00～15:30 |

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

Abstract | Although rotational surfaces in L3 have been widely considered in the literature, its complete classification has been avoided because of its difficulty. In the firs part of this talk a complete classification of rotational surfaces in L3 will be given. We will focus specially in the case with space-like axis, which is the most interesting and subtle. The method to study this surfaces is delicate and intricate, and it is based in a technique of surgery and gluing. This method will be illustrated with an algorithm to construct new examples. In the second part, we will obtain the rotational Willmore surfaces in L3. |

Speaker | Mamoru Doi (Osaka Univ., Dept. of Math.) |
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Title | Gluing construction of manifolds with a special closed differential form and its applications |

Date | July 22 (Tue.) 14:40～16:10 |

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

Abstract | Complex manifolds with trivial canonical bundle and Riemannian manifolds with vanishing Ricci curvature are examples of manifolds whose geometric structure is characterized by a special closed differential form. In this talk, we will introduce a gluing method for constructing manifolds such as complex surfaces with trivial canonical bundle, Spin(7) manifolds, etc., and discuss its applications. |

Speaker | Yoshihiro OHNITA (Osaka City Univ. ) |
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Title | On Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces |

Date | July 16 (Wed.) 14:40～16:10 |

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

Abstract | Hypersurfaces with constant principal curvatures in real space forms are called " isoparametric hypersurfaces " in differential geometry . The theory of isoparametric hypersurfaces has long history since Elie Cartan and is actively investigated even now. By the famous result of Muenzner, it is known that the number g of distinct principal curvatures of isoparametric hypersurfaces in spheres must be g=1,2,3,4,6. The complex hyperquadric is a typical one of compact Hermitian symmetric spaces of rank 2. There is known to be the relationship between hypersurface geometry in a sphere and Lagrangian submanifolds in a complex hyperquadrics. The images of the Gauss maps of isoparametric hypersurfaces in spheres provide a nice class of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics. In this talk we will discuss the Hamiltonian stability of compact minimal Lagrangian submanifolds obtained as the Guuss images of homogeneous isoparametric hypersurfaces. Moreover we will mention our recent results on the problem in my joint work with Dr. Hui Ma (Tsinghua University, Peking) and especially Hamiltonian stable and Hamiltonian unstable examples in case g=4. |

Speaker | Yasuyuki NAGATOMO ( Dept. of Math., Kyushu Univ. ) |
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Title | Totally geodesic submanifolds of Grassmann manifolds |

Date | July 9 (Wed.) 14:40～16:10 |

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

Abstract | We define a totally geodesic submanifold of irreducible type of a Grassmannian manifold. The classification of such submanifolds will be discussed in a seminar. Then we obtain an integral formula which determines the dimension of the ambient space. We use a similar theory to a theory of spherical functions to show that any full totally geodesic submanifold of imdecomposable type is of irreducible type, in the case that the domain is a complex projective line. Thus, we determine all totally geodesic submanifold from a complex projective line into Grassmannian. |

Speaker | Takashi SUZUKI ( Dept. of Informatics and Mathematical Science, Osaka Univ. ) |
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Title | Recent topics in diffusion geometry and nonlinear analysis -harmonic flow, chemotaxis system, normalized Ricci flow, tumour growth model |

Date | June 18 (Wed.) 14:40～16:10 |

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

Abstract | In spite of several differences in the methods of analysis and expected phenomena, we can observe common philosophy in the study of nonlinear problems in diffusion geometry and nonlinear analysis. Motivated by recent analytic result on the normalized Ricci flow, I illsutrate the mass quantization in the Smoluchowski-Poisson equation, and describe the similarity and the difference between the harmonic heat flow, normalized Ricci flow, and cancer model, and also directly related stationary problems in the self-dual gauge field and the mean field turbulence. |

Speaker | Takayuki MORIYAMA ( Dept. of Math., Osaka Univ. ) |
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Title | Deformations of transverse Calabi-Yau structures on foliated manifolds |

Date | June 11 (Wed.) 13:00～14:30 |

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

Abstract | We develop a deformation theory of transverse structures given by closed forms on foliated manifolds. We apply Goto's deformation theory to transverse structures on foliated manifolds and show that a deformation space of the transverse structures is smooth under a cohomological condition. As an application, we obtain unobstructed deformations of transverse Calabi-Yau structures on foliated manifolds. |

Speaker | Hironori Sakai (Tokyo Metropolitan University, D3) |
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Title | Restoration of quantum orbifold cohomology from quantum D-modules |

Date | June 4 (Wed.) 14:40～16:10 |

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

Abstract | We study the D-module associated to the quantum orbifold cohomology of a weighted projective space (quantum D-module). We see how to restore the quantum orbifold cohomology from a quotient description of the quantum D-module. (Joint work with Martin Guest.). |

Speaker | Ken-ichi Sakan (Osaka City University) |
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Title | On asymptotically sharp inequalities for quasi conformal harmonic mappings |

Date | May 21 (Wed.) 14:40～16:10 |

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

Abstract | Japanese page only |

Speaker | Prof. Oldrich Kowalski （Charles University in Prague, The Czech Republic） |
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Title | On 3-dimensional Riemannian manifolds with prescribed Ricci eigenvalues |

Date | May 14 (Wed.) 14:40～16:10 |

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

Abstract | Japanese page only |

Speaker | Mr. Kota Hattori (Univ. of Tokyo, D2) |
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Title | A rigidity theorem for quaternionic Kaehler structures |

Date | May 7 (Wed.) 16:30～18:00 |

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

Abstract | We can construct the twistor theory on quaternionic Kaehler manifolds which are kind of Riemannian manifolds with special holonomy group. So they are the interesting objects of study for both Riemannian geometry and complex geometry. The rigidity theorems for quaternionic Kaehler structures has already been proved by LeBrun, Salamon, and Horan. In this talk, we will prove this type of theorem by using Riemannian geometry, without twistor theory. To prove it, we use the topological calibration theory by Ryushi Goto and Bochner-Weitzenbock formula for the quaternionic Kaehler manifolds by Yasushi Homma. |

Speaker | Professor Akira Asada |
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Title | Regularized infinite dimensional integral and Fourier expansion on infinite dimensional torus |

Date | April 6 (Wed.) 14:40～16:10 |

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

Abstract | Fixing an operator G, regularized infinite dimensional integral is defined by using zeta regularization of eigen values of G. As an application of regularized infinite dimensional integral, we compute Fourier expansion of periodic functionof infinite variables (functions on infinite dimensional torus). In this case, periods are not arbitrary, but restricted by the eigen values of G. The lattice generated by periods should be considered not in a Hilbert space, but in a Hilbert space added a 1-diemnsioanl space (Hilbert space added the determinant bundle). Fourier expansion on the torus need to use not only finite product of trigonometric functions, but also infinite product of trigonometric functions. Relation of these results and periodic boundary value problem of regularized Lapalacian on a Hilbert space is also explained, if possible. |