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過去の記録 ～04/15｜次回の予定｜今後の予定 04/16～
| 開催情報 | 金曜日 10:00～11:30 数理科学研究科棟(駒場) 126号室 |
|---|---|
| 担当者 | 植田一石，金井雅彦，二木昭人 |
| 備考 | 開始時間と開催場所などは変更されることがあるので, セミナーごとにご確認ください. |
2016年04月21日(木)
17:00-18:00 数理科学研究科棟(駒場) 123号室
集中講義に続いて行います.いつもと違う部屋ですのでご注意下さい.
本多正平 氏 (東北大学)
Spectral convergence under bounded Ricci curvature (Japanese)
集中講義に続いて行います.いつもと違う部屋ですのでご注意下さい.
本多正平 氏 (東北大学)
Spectral convergence under bounded Ricci curvature (Japanese)
[ 講演概要 ]
For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are sharp generalizations of Cheeger-Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. These spectral convergence have two direct corollaries. One of them is to give new bounds on such eigenvalues, in terms of bounds on volume, diameter and the Ricci curvature. The other is that we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. On the other hand we also define measurable curvature tensors of the noncollapsed Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a uniform bound of Ricci curvature, which include Riemannian curvature tensor, the Ricci curvature, and the scalar curvature. As fundamental properties of our Ricci curvature, we show that the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover we prove a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology by using our scalar curvature. This talk is based on arXiv:1510.05349.
For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are sharp generalizations of Cheeger-Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. These spectral convergence have two direct corollaries. One of them is to give new bounds on such eigenvalues, in terms of bounds on volume, diameter and the Ricci curvature. The other is that we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. On the other hand we also define measurable curvature tensors of the noncollapsed Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a uniform bound of Ricci curvature, which include Riemannian curvature tensor, the Ricci curvature, and the scalar curvature. As fundamental properties of our Ricci curvature, we show that the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover we prove a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology by using our scalar curvature. This talk is based on arXiv:1510.05349.
