Development of Integrable Geometry
Japan Society for the Promotion of Science
Grants-in-Aid for Scientific Research
Miyaoka Reiko, KOTANI Motoko, NISHINOU Takeo, UEHARA Taketo, MATSUURA Nozomu, IWASAKI Katsunori, IRITANI Hiroshi, KAJIWARA Kenji, NAGATOMO Yasuyuki, NOMURA Takaaki, YAMADA Kotaro, ISHIKAWA Goo, UMEHARA Masaaki, GUEST Martin, SHODA Toshihiro, FUTAKI Akito, FUJIOKA Atsushi, RASSMAN Wayne, TAMARU Hiroshi
4 2011
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3 2015
Grant number：23340012
Grant amount：\13780000
（
Direct Cost: \10600000
、
Indirect Cost：\3180000
）
Isoparametric hypersurfaces with 6 principal curvatures with multiplicity 2 are shown to be homogeneous, which solves one of Yau's problems. As for 4 principal curvature case, we gave a description by using the moment map of spin actions. Transnormal systems are investigated in details.
We show the non-existence of L2 harmonic 1-form on a complete non-compact stable minimal Lagrangian submanifolds in a Kaheler manifold with positive Ricci curvature. Then the number of non-parabolic ends is less than two, and in the surface case, the genus should vanish. The Floer theory on the intersection of a Lagrangian submanifold with its Hamiltonian deformation is investigated. The Gauss images of isoparametric hypersurfaces in the sphere are Lagrangian submanifolds of complex hyperquadric, and in this case, we show that if the multiplicities of the principal curvatures are bigger than 1, then they are Hamiltonian non-displaceable,
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Fusion of geometry and the theory of integrable systems
Japan Society for the Promotion of Science
Grants-in-Aid for Scientific Research
MIYAOKA Reiko, OHNITA Yoshihiro, MOTOKO Kotani, SASAKI Takeshi, IWASAKI Katsunori, OYSU Yukio, KAJIWARA Kenji, NAGATOMO Yasuyuki, NAKAYAAHIKI Atsushi, YAMADA Kotaro, FUTAKI Akito, MARTIN Guest, WAYNE Rossman, SHODA Toshihiro, IRITANI Hiroshi, ISHIKAWA Goo, UMEHARA Masaaki, KAWAKUBO Satoshi, TAMARU Hiroshi, FUJIOKA Atsushi, MATSUURA Nozomu, NISHINOU Takeo
2007
-
2010
Grant number：19204006
Grant amount：\27560000
（
Direct Cost: \21200000
、
Indirect Cost：\6360000
）
We classified almost all isoparametric hypersurface, and characterize them in terms of the moment map, which proves the evidence of a relation with integrable systems. A basic theory of surfaces with singularities, and a new method using the Legendre map have been established. Via the Riemann-Hilbert correspondence, the dynamical system of Painleve equations is investigated, and the view point of the chaos has been developed. The modularity of higher genus Gromov-Witten and the mirror symmetry are discussed. A surface with potential appeared in quantum cohomology is constructed, which contributes to the tt* geometry.
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