发表机构
Simons Center for Geometry and Physics, Stony Brook University; Institute for Advanced Study in Mathematics, Zhejiang University（石溪大学西蒙斯几何与物理中心; 浙江大学数学高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文发展了紧致单连通环面爱因斯坦四流形的系统理论，通过局部粘合与虚拟计数构造任意大$b_2$的正爱因斯坦度量，并给出代数特殊度量的拓扑刻画、唯一性及微分同胚分类。
AI 中文摘要
本文是我们之前关于环面引力瞬子论文的延续。我们发展了具有环面对称性的紧致单连通爱因斯坦四流形的系统理论。应用包括：（1）构造具有任意大$b_2$的单连通正爱因斯坦四流形。我们的方法基于局部粘合与全局虚拟计数论证相结合。关键思想涉及杆结构的特殊设计，以实现粘合构造并限制可能的退化。（2）代数特殊环面爱因斯坦度量的拓扑刻画。这利用了环面背景下$W^+$的特殊曲率恒等式。（3）在允许代数特殊爱因斯坦度量的环面四流形上，在缩放和等距意义下的唯一性。证明使用了爱因斯坦-希尔伯特泛函的变分研究。（4）基于改进的Hitchin-Thorpe不等式$\chi\geq3|\tau|$的紧致环面爱因斯坦四流形的微分同胚分类。特别地，我们的构造从圆球出发，通过锥形爱因斯坦度量的模空间，恢复了所有经典的环面爱因斯坦度量，包括Page和Chen--LeBrun--Weber度量。
英文摘要
This is a continuation of our previous paper on toric gravitational instantons. We develop a systematic theory of compact simply-connected Einstein four-manifolds with toric symmetry. Applications include (1) The construction of simply connected positive Einstein four-manifolds with arbitrarily large $b_2$. Our method is based on local gluing combined with a global virtual counting argument. The key idea involves a particular design of rod structures to enable a gluing construction and restrict possible degenerations. (2) A topological characterization of algebraically special toric Einstein metrics. This uses special curvature identities for $W^+$ in the toric setting. (3) Uniqueness, up to scaling and isometry, on toric four-manifolds admitting algebraically special Einstein metrics. The proofs uses a variational study of the Einstein--Hilbert functional. (4) Diffeomorphism classification of compact toric Einstein four-manifolds in terms of an improved Hitchin-Thorpe inequality $χ\geq3|τ|$. In particular, our construction recovers all the classical toric Einstein metrics, including the Page and Chen--LeBrun--Weber metrics, starting from the round sphere through moduli spaces of conical Einstein metrics.
Comments136 pages. Comment welcome
