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Lie Groups and Representation Theory Seminar

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Date: July 28 (Wed), 2021, 17:00-18:00
Speaker: Yoshiki Ohima (大島芳樹) (Osaka University)
Title: Ricci平坦計量の崩壊とMonge-Ampere方程式の解のアプリオリ評価 / Collapsing Ricci-flat metrics and a priori estimate for the Monge-Ampere equation
[ pdf ]
YauはMonge-Ampere方程式の解のアプリオリ評価を行ってCalabi予想を証明した. 近年ファイバー空間の構造を持つCalabi-Yau多様体 について,底空間のKahler類に崩壊するようなRicci平坦Kahler計量の振舞が Gross-Tosatti-Zhang等により研究されている.尾高悠志 との共同研究(arXiv:1810.07685)で得られたK3曲面の球面へのGromov-Hausdorff収束も,これらのMonge-Ampere方程式の解の評価に基 づいている.この講演では,微分方程式の解の評価がどのように自然な計量の存 在やGromov-Hausdorff収束を導くかをお話ししたい.

Yau proved the Calabi conjecture by using a priori estimate for the Monge-Ampere equation. Recently, for a Calabi-Yau manifold with a fiber space structure, the behavior of Ricci-flat metrics collapsing to a Kahler class of the base space was studied by Gross-Tosatti-Zhang, etc. The Gromov-Hausdorff convergence of K3 surfaces to spheres obtained by a joint work with Yuji Odaka (arXiv:1810.07685) is also based on those estimates for solutions to the Monge-Ampere equation. In this talk, I would like to discuss how an estimate of solutions to differential equations deduces the existence of canonical metrics and the Gromov-Hausdorff convergence.

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