複素解析幾何セミナー

過去の記録 ~12/02次回の予定今後の予定 12/03~

開催情報 月曜日 10:30~12:00 数理科学研究科棟(駒場) 128号室
担当者 平地 健吾, 高山 茂晴, 野村 亮介

次回の予定

2021年12月13日(月)

10:30-12:00   オンライン開催
川村 昌也 氏 (香川高等専門学校)
A generalized Hermitian curvature flow on almost Hermitian manifolds (Japanese)
[ 講演概要 ]
It is well-known that the Uniformization theorem (any Riemannian metric on a closed 2-manifold is conformal to one of constant curvature) can be proven by using the Ricci flow. J. Streets and G. Tian questioned whether or not a geometric flow can be used to classify non-Kähler complex surfaces as in the case of the Ricci flow. Also they asked if it is possible to prove classification results in higher dimensions by using geometric flows in non-Kähler Hermitian geometry. Streets and Tian considered that these flows should be close to the Kähler-Ricci flow as much as possible. From this point of view, they introduced a geometric flow called the Hermitian curvature flow (HCF) which evolves an initial Hermitian metric in the direction of a Ricci-type tensor of the Chern connection modified with some lower order torsion terms. Streets and Tian also introduced another geometric flow, which is called the pluriclosed flow (PCF), by choosing torsion terms to preserve the pluriclosed condition on Hermitian metrics. Y. Ustinovskiy studied a particular version of the HCF over a compact Hermitian manifold. Ustinovskiy proved that if the initial metric has Griffiths positive (non-negative) Chern curvature, then this property is preserved along the flow.

In recent years, some results concerning geometric flows on complex manifolds have been extended to the almost complex setting. For instance, L. Vezzoni defined a new Hermitian curvature flow on almost Hermitian manifolds for generalizing some studies on the HCF and the Hermitian Hilbert functional. And J. Chu, V. Tosatti and B. Weinkove considered parabolic Monge-Ampère equation on almost Hermitian manifolds, which is equivalent to the almost complex Chern-Ricci flow. T. Zheng characterized the maximal existence time for a solution to the almost complex Chern-Ricci flow.

In this talk, we consider a generalized Hermitian curvature flow in almost Hermitian geometry and introduce that it has some properties such as the long-time existence obstruction, the uniform equivalence between its solution and an almost Hermitian metric, and the preservation result along the flow.
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB