Positive Einstein metrics on infinitely many four-manifolds

Abstract

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 $2\pi\beta$ of a prescribed singularity along a torus, starting from the product metric on $\mathbb S^2\times\mathbb S^2$ and from the symmetric K\"ahler-Einstein metric on $\mathbb{CP}^2\#3\overline{\mathbb{CP}}{}^2$. At $\beta=\frac{2}{k+1}$, 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.