无穷多个四流形上的正爱因斯坦度量
Positive Einstein metrics on infinitely many four-manifolds
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中文总结 AI 辅助
该论文在两类四流形上构造了正标量曲率的光滑爱因斯坦度量，证明无穷多个同胚类型存在此类度量，并通过锥角变化实现拓扑改变，最终坍缩至二维圆盘。
中文摘要 AI 辅助
我们在 $\\#_k({\mathbb S^2}\times {\mathbb S^2})$ 和 $k\mathbb{CP}^2\\#(2k+1)\overline{\mathbb{CP}}{}^{2}$ 上构造了具有正标量曲率的光滑爱因斯坦度量，其中 $k\geqslant2$。特别地，我们证明了在无穷多个闭 $4$-流形的同胚类型上存在正爱因斯坦度量。第一个族表明，任何承载正标量曲率度量的单连通自旋 $4$-流形都同胚于一个爱因斯坦 $4$-流形。Wick 旋转给出了相关的洛伦兹静态轴对称爱因斯坦度量，具有正宇宙学常数和任意多个具有相同表面引力的视界。这些度量通过改变沿环面的指定奇点的锥角 $2\pi\beta$ 获得，从 $\mathbb S^2\times\mathbb S^2$ 上的乘积度量以及 $\mathbb{CP}^2\\#3\overline{\mathbb{CP}}{}^2$ 上的对称 Kähler-Einstein 度量出发。在 $\beta=\frac{2}{k+1}$ 时，改变环面周围角度的周期会移除锥奇点并改变拓扑。对于两个族，当 $k\to+\infty$ 时，度量沿环面轨道坍缩到一个显式的 $2$-维圆盘，其边界的 Hausdorff 维数为 $\frac{4}{3}$。在曲率尺度下，它们收敛到具有 Kasner 渐近性的完备周期 Ricci 平坦度量：第一个族为欧几里得 Myers-Korotkin-Nicolai 度量，第二个族为一个新的度量。
英文摘要
We construct smooth Einstein metrics with positive scalar curvature on $\#_k({\mathbb S^2}\times {\mathbb S^2})$ and on $k\mathbb{CP}^2\#(2k+1)\overline{\mathbb{CP}}{}^{2}$, for $k\geqslant2$. In particular, we show that positive Einstein metrics exist on infinitely many homeomorphism types of closed $4$-manifolds. The first family shows that any simply connected spin $4$-manifold carrying metrics with positive scalar curvature is homeomorphic to an Einstein $4$-manifold. Wick rotation gives associated Lorentzian static axisymmetric Einstein metrics with positive cosmological constant and arbitrarily many horizons with the same surface gravity. The metrics are obtained by varying the cone angle $2πβ$ of a prescribed singularity along a torus, starting from the product metric on $\mathbb S^2\times\mathbb S^2$ and from the symmetric Kähler-Einstein metric on $\mathbb{CP}^2\#3\overline{\mathbb{CP}}{}^2$. At $β=\frac{2}{k+1}$, changing the period of the angle around the torus removes the cone singularity and changes the topology. For both families, as $k\to+\infty$, the metrics collapse along the torus orbits to an explicit $2$-dimensional disk, whose boundary has Hausdorff dimension $\frac{4}{3}$. At the scale of curvature, they converge to complete periodic Ricci-flat metrics with Kasner asymptotics: the Euclidean Myers-Korotkin-Nicolai metric for the first family, and a new one for the second.
