# Differential Geometry Seminar (2013)

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 Kaname Hashimoto 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 h-kaname@sci.osaka-cu.ac.jp |

Speaker | Shouhei Honda （Kyushu Univ.） |
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Title | Gromov-Hausdorff収束版Rellich型コンパクト性定理とその応用 |

Date | March 19 (Wed.) 2014, 14:40～16:10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | Japanese page only |

Speaker | Hitoshi Yamanaka（OCAMI） |
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Title | Representation coverings and equivariant hyperbolic diffeomorphisms |

Date | March 3 (Mon.) 2014, 14:40～16:10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | The existence of a Morse function on a closed manifold is widely known. On the other hand, any general existence theorem for invariant Morse functions is not known except a result of Wasserman concering the existence of invariant Bott-Morse fucntions. In this talk, I will introduce the notion of a representation covering, and show that the existence of a certain equivariant hyperbolic diffeomorphism implies the existence of a representation covering. As a corollary, one finds that the existence of a representsation covering gives an obstruction for the existence of an invariant Morse function. Moreover, as a converse of this result, I will show that in the case of a certain holomorphic torus action, the existence of a representation covering implies the existence of an equivariant hyperbolic diffeomorpshism. |

Speaker | Takahiro Noda （Nagoya University & OCAMI） |
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Title | Contact geometry of partial differential equations |

Date | February 19 (Wed.) 2014, 14:40～16:10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | In this talk, we explain contact geometry of partial differential equations. Especially, we investigate partial differential equations with one dependent variables and exhibit the rich examples. |

Speaker | Hiroshi Irie (Tokyo Denki Univ.) |
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Title | Oh の Hamilton 体積最小性予想についての二つの注意 |

Date | February 13 (Thu.) 2014, 10：40～12：10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | Japanese page only |

Speaker | Sorin V. SABAU (Tokai University, Sapporo Campus) |
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Title | Some convexity related problems in Finsler geometry |

Date | January 30 (Thu.) 2014, 16:30～18:00 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | The notion of convexity radius was introduced by J. C. H. Whitehead, but the existence and basic properties were studied only in the analytical and the absolute homogeneous case. We will show the existence of convexity radius for an arbitrary Finsler metric and we study the relation with the injectivity radius. Moreover, we discuss geodesic coordinates and other related topics. |

Speaker | Kota Hattori (Univ. of Tokyo) |
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Title | Taub-NUT変形の一般化 |

Date | March 19 (Wed.) 2014, 14:40～16:10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | Japanese page only |

Speaker | Kentaro Saji (Kobe University) |
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Title | Geometric invariants of cuspidal edge |

Date | Dec. 18 (Thu.) 2013, 14:40～16:10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | A cuspidal edge is a germ which can be transformed to (u, v^2, v^3) by diffeomorphism-germs on the source and the target. We introduce several geometric invariants of curpidal edges and study their properties. |

Speaker | Atsuhide Mori (OCAMI) |
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Title | On regularizations of b-Poisson structures |

Date | Dec. 4 (Thu.) 2013, 14：40 ～ 16：10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | Topology of contact structures have close relationship with that of corank one regular Poisson structures. However, we do not know much about either of them. A b-Poisson structure is the most tractible non-regular Poisson structure. Given it on a 2n-manifold, we can construct a certain corank one regular Poisson 2n+1-manifold. Marcut and Osorno, in arXiv:1303.6246, pointed out that this construction is no use for many interesting cases, e.g. S^4 S^1. I will give a little ingenuity on this topic. |

Speaker | Tetsu Toyoda（Suzuka National College of Technology） |
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Title | Convexity of Rayleigh quotients for nonlinear spectral gaps and optimal metrics on graphs |

Date | Nov. 27 (Thu.) 2013, 14：40 ～ 16：10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | In this talk, we study the existence and symmetry of optimal metrics on weighted graphs with respect to nonlinear spectral gaps. We show that on any finite connected weighted graph, there exists an optimal metric which carries the symmetry of the weighted graph. The key is to establish the convexity of ``Rayleigh quotients" in an appropriate setting. |

Speaker | Takumi Yamada（Shimane University） |
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Title | 旗多様体上の不定値ケーラー計量の指数と不変複素構造について |

Date | Nov. 13 (Wed.) 2013, 14：40～16：10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | It is well known that a pseudo-Kaehler structure is one of the natural Generalizations of a Kaehler structure. In this talk, we consider signatures of invariant pseudo-Kaehler metrics on generalized flag manifolds from the Viewpoint of T-root system. |

Speaker | Atsufumi Honda（Miyakonojo National College of Technology） |
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Title | 凸でない非等方的密度関数とMinkowski空間のCMC曲面 |

Date | Oct. 23 (Wed.) 2013 14：40～16：10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | We call surfaces which are in equilibrium for a constant coefficient parametric functional with a volume constraint CAMC. Usually, they are imposed so-called ``convexity condition''. In this talk, we introduce some results about CAMC surfaces with nondegenerate condition. |

Speaker | Hajime Urakawa（Institute for International Education, Tohoku University） |
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Title | Toward B.Y. Chen's conjecture on biharmonic immersions |

Date | Oct. 16 (Wed.) 2013 14：40～16：10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | B.Y. Chen's conjecture on biharmonic immersions is that the only biharmonic submanifolds of the Euclidean space might be minimal. One of methods to construct submanifolds of the Euclidean space is to consider Lagrangian submanifolds L of a symplectic manifold N, especially, to take a graph Graph(u) of a closed 1-form u on M in the cotangent bundle N=T*M. We will discuss when Graph(u) is biharmonic or minimal. We would like to approach B.Y. Chen's conjecture. |

Speaker | Jason Lotay (Univ.College London) |
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Title | (1) Coassociative conifolds 1: smoothings of cones(2) Coassociative conifolds 2: singularities and stability |

Date | August 12 (Mon.) 2013, (1) 1:30-2:30 (2) 4:00-5:00 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | (1) 1:30-2:30 Coassociative conifolds 1: smoothings of conesCoassociative 4-folds are important examples of calibrated, hence volume-minimizing, submanifolds and are inherently related to Riemannian manifolds with exceptional holonomy group G_2. In this first talk, I will discuss the theory of asymptotically conical coassociative 4-folds, which are smoothings of coassociative cones, including describing their moduli space of deformations. These submanifolds are particularly important for providing local models for resolving singular coassociative 4-folds.<\br> (2) 4:00-5:00 Coassociative conifolds 2: singularities and stability<\br> Singular coassociative 4-folds help us to understand the boundary of the moduli space of smooth coassociative 4-folds and are important from the point of view coassociative fibrations of compact G_2 manifolds. One of the simplest models of a singularity is given by a cone. In this second talk, I will discuss the theory of coassociative conical singularities, with a particular focus on the role of a numerical invariant associated to coassociative cones called the stability index. |

Speaker | Masashi Ishida（Osaka University） |
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Title | Diameter bounds in geometric flows |

Date | July 24 (Wed.) 2013, 14：40～16：10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | An interesting problem on geometric flow is whether the diameters of manifolds under the evolving metrics stay bounded. Perelman proved that the diameter stays bounded for the Kahler-Ricci flow on Fano manifolds. For a general Ricci flow, in 2005, Peter Topping proved an upper bound of the diameter by applying the monotonicity of Perelman's W-entropy functional. In this talk, I would like to discuss the problem for a more general geometric flow including the Ricci flow as a special case. If the time permits, I also would like to mention an attempt to generalize a recent work of Qi S. Zhang (July, 2013) who refined the result of Topping. |

Speaker | Hiroaki Ishida（JSPS PD, RIMS） |
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Title | Complex manifolds with maximal torus actions |

Date | July 17 (Wed.) 2013, 14：40～16：10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | Let G be a compact torus acting on a connected manifold M effectively. We say that the G-action on M is maximal if there exists a point x in M such that dim G+dim G_x = dim M. In this talk, I would like to explain a classification of complex manifolds with maximal torus actions. |

Speaker | Yu Kawakami (Yamaguchi University) |
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Title | The purpose of this talk is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms (minimal surfaces in the Euclidean 3-space, improper affine spheres in the affine 3-space and etc.). For this purpose, we give an effective curvature bound for a specified conformal metric on an open Riemann surface. |

Date | July 12 (Fri.) 2013 15：00～16：30 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | Japanese page only |

Speaker | Masaru Hasegawa (Saitama University) |
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Title | A Koenderink type theorem for Whitney umbrellas |

Date | July 3 (Wed.) 2013, 14：40～16：10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | Koenderink showed that for regular surfaces in 3-dimensional Euclidean space, the Gaussian curvature of the surface at a given point is the product of the curvature of the contour viewed in a direction with the normal curvature in the direction. In this talk, we introduce a Koenderink type theorem for Whitney umbrellas. This talk is based on the joint work with Toshizumi Fukui (Saitama) and Kentaro Saji (Kobe). |

Speaker | Osamu Kobayashi (Osaka University) |
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Title | The Yamabe invariant in dimension four |

Date | June 19 (Wed.) 2013 14：40～16：10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | In 1990's the theory of the Yamabe invariant made great progress mainly due to C. LeBrun. About15 years have passed since then. There seems however few remarkable progress. In this talk I review LeBrun's work, and explain the present state about the Yamabe invariant in dimension four. |

Speaker | Yukinori YASUI (Department of Physics, Osaka City University) |
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Title | CKY symmetry and deformation of Sasakian Structure |

Date | May 29 (Wed.) 2013 14：40～16：10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | It is known that conformal Killing-Yano (CKY) tensors describe a ``hidden symmetry " of the Kerr black hole. The CKY symmetries play an important role in the study of classification of black holes and the stability analyses of spacetimes. We first review our research of the CKY according to the paper, Prog.Theor.Phys. Supple. 189 (2011). Next we discuss a deformation of the CKY symmetry by using a torsion of spacetime. This is based on the recent paper arXiv:1207.0247. In the fundamental equations of supergravity theories there exists a 3-form field interacting with a string. Such a field can be naturally interpreted as a torsion of spacetime. Especially, deformation of Calabi-Yau and Sasakian structure in the presence of torsion is important since these geometries describe the supersymmetric solutions of supergravity theories. It is shown that the CKY symmetries work well for the explicit construction of the Calabi-Yau and Sasaki manifolds with torsion. |

Speaker | Dr. Makiko Make (Tokyo Metropolitan University/OCAMI) |
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Title | Correspondence among families of certain K3 surfaces |

Date | April 9 (Tue.) 2013, 14：40～16：10 |

Place | Dept. of Mathematics, General Research Bldg., 301 |

Abstract | The Picard lattices of families of weighted K3 surfaces are computed by Belcastro and it is observed that some of these lattices are isometric. In the first part, we prove that if the Picard lattices of families of weighted K3 surfaces are isometric, then, the general members in the families are birationally correspondent. In the second part, we discuss about the families of K3 surfaces in smooth toric Fano 3-folds. We can compute the Picard lattices of general members in the families, and prove that these lattices are mutually distinct. Therefore, unlike the first part, we can prove by Torelli-type theorem that there is no birational correspondence among the families of K3 surfaces in smooth toric Fano 3-folds. In the third part, we introduce certain two families of K3 surfaces in smooth Fano 3-folds and discuss a birational correspondence between the members in these families. Finally we present some proceeding study related to bimodal singularities. |