紧致李群上的广义数量曲率与Ricci流

中文摘要 AI 辅助

设$G$为紧致连通半单李群，$Q$为双不变度量，$\rho$为其Cartan三形式。我们证明，满足$[H_0]=[\rho]$的每个左不变广义Ricci流对所有正时间均存在，并在无需缩放或拉回的情况下光滑收敛到Bismut-Ricci平坦对。标准对在其挠类中唯一地最大化广义数量曲率，且紧致单群上的每个左不变Bismut-Ricci平坦对都是标准的，这回答了Fusi、Lafuente和Stanfield以及Lauret和Will的问题。在紧致单群的两个拷贝的乘积上，Cartan类中的非标准驻点对是已知的；我们构造了具有精确初始形式$[H_0]=[\rho]$和非标准极限的非驻点流，从而否证了Garcia-Fernandez和Streets收敛猜想的一般半单情形。证明使用了恰当的非增能量以及将驻点对参数化为幂零李代数自同态的Killing场的分解。

英文摘要

Let $G$ be a compact connected semisimple Lie group, let $Q$ be a bi-invariant metric and let $ρ$ be its Cartan three-form. We prove that every left-invariant generalized Ricci flow with $[H_0]=[ρ]$ exists for all positive time and converges smoothly to a Bismut-Ricci-flat pair without rescaling or pullback. The standard pair uniquely maximizes generalized scalar curvature in its torsion class, and every left-invariant Bismut-Ricci-flat pair on a compact simple group is standard, answering questions of Fusi, Lafuente and Stanfield and of Lauret and Will. On the product of two copies of a compact simple group, nonstandard stationary pairs in the Cartan class are known; we construct nonstationary flows with the exact initial form $[H_0]=[ρ]$ and nonstandard limits, disproving the general semisimple case of the convergence conjecture of Garcia-Fernandez and Streets. The proofs use a proper nonincreasing energy and a decomposition of Killing fields that parametrizes the stationary pairs by nilpotent Lie algebra endomorphisms.

发表机构

• School of Mathematics and Statistics, Guangdong University of Technology（广东工业大学数学与统计学院）
• School of Mathematical Sciences, South China Normal University（华南师范大学数学科学学院）
• Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University（南京师范大学数学科学学院数学研究所）

机构由 AI 辅助整理，请以论文原文为准。