论文标题
关于超声流动流的报告
A report on the hypersymplectic flow
论文作者
论文摘要
本文讨论了一个相对较新的几何流量，称为超声流动。在本文的上半年，我们解释了该流程的原始动机思想，来自4维符号拓扑和7维$ G_2 $ - 几何。我们还调查了流量的最新进展，最著名的是延伸定理，假设有标量曲率结合。下半场包含新的结果。我们证明，完全无扭转的超透明结构必须是Hyperkähler。我们表明，涉及标量曲率的某种积分结合可以排除超圆连流中有限的时间奇异性。我们表明，如果初始的超透明结构足够接近点正交，则该流程一直存在。最后，我们证明了在一些强有力的假设下流动的融合，包括长期存在。
This article discusses a relatively new geometric flow, called the hypersymplectic flow. In the first half of the article we explain the original motivating ideas for the flow, coming from both 4-dimensional symplectic topology and 7-dimensional $G_2$-geometry. We also survey recent progress on the flow, most notably an extension theorem assuming a bound on scalar curvature. The second half contains new results. We prove that a complete torsion-free hypersymplectic structure must be hyperkähler. We show that a certain integral bound involving scalar curvature rules out a finite time singularity in the hypersymplectic flow. We show that if the initial hypersymplectic structure is sufficiently close to being point-wise orthogonal then the flow exists for all time. Finally, we prove convergence of the flow under some strong assumptions including, amongst other things, long time existence.
