厄米特曲率流与HKT几何
Hermitian curvature flow and HKT geometry
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中文总结 AI 辅助
该研究确定了保持HKT几何的厄米特曲率流，证明了四维情形下其极大存在时间猜想，发现四元数Hopf曲面上的发散HKT-爱因斯坦度量序列，并分类了李群双不变度量对应的强HKT结构。
中文摘要 AI 辅助
我们确定了一种保持HKT几何的厄米特曲率流，其不动点为HKT-爱因斯坦度量，等价于Verbitsky在四元数Monge-Ampère方程背景下提出的流。我们给出了陈标量曲率的基本正则性障碍与单调性公式，对该流提出了极大存在时间猜想并给出了条件性解决，在四维情形证明了该存在性猜想。我们证明了四元数Hopf曲面上存在发散的HKT-爱因斯坦度量序列，这些是文献中首批非齐次例子，表明收敛问题的复杂性。最后我们对李群双不变度量产生的强HKT结构中哪些同时为HKT-爱因斯坦结构进行了分类。
英文摘要
We identify a Hermitian curvature flow which preserves HKT geometry, and whose fixed points are HKT-Einstein metrics, equivalent to a flow suggested by Verbitsky in the context of the quaternionic Monge-Ampère equation. We exhibit a fundamental regularity obstruction and a monotonicity formula for the Chern scalar curvature. We formulate a maximal existence time conjecture for this flow, and give a conditional resolution. We establish the existence conjecture in dimension four. We show the existence of a divergent sequence of HKT-Einstein metrics on quaternionic Hopf surfaces. These are the first non-homogeneous examples in the literature, and indicate the delicacy of the convergence question. Finally we classify which strong HKT structures arising from bi-invariant metrics on Lie groups are also HKT-Einstein.
