Latest work
Compact toric Einstein four-manifolds
Our latest work gives a complete diffeomorphism classification of compact, simply connected four-manifolds admitting positive toric Einstein metrics. The proof combines a detailed study of moduli spaces of conical Einstein metrics with a global boundary-counting argument.
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A closed, simply connected smooth four-manifold $X$ admitting an effective $T^2$-action carries a positive toric Einstein metric if and only if the inequality above holds. Here $\chi$ is the Euler characteristic and $\tau$ is the signature. The torus action preserving the metric may differ from the initially given action. In particular, we construct positive Einstein metrics on every connected sum $\#_r(S^2\times S^2)$, giving examples with arbitrarily large second Betti number.
The main idea is to allow conical singularities along selected torus-invariant two-spheres. Varying their common cone angle $2\pi\beta$ gives a moduli problem of virtual dimension one. We construct ends of this space by gluing conical, asymptotically Euclidean gravitational instantons into known compact Einstein manifolds at torus fixed points. Carefully chosen torus actions and compactness estimates control the possible degenerations. A virtual boundary-counting argument then forces the existence of metrics with cone angle $2\pi$, where the singularities disappear. This approach is inspired by moduli-space methods in Donaldson theory.
We also establish rigidity when $b_+\leq 1$: for a fixed orientation, a toric Einstein metric is anti-self-dual when $b_+=0$ and conformally Kähler when $b_+=1$. Together with classical classification results, this gives uniqueness of toric Einstein metrics up to scaling and isometry on $S^4$, $S^2\times S^2$, and $\CP^2\#k\overline{\CP}^{2}$ for $0\leq k\leq 3$. Our construction recovers all six classical toric Einstein metrics, including the Page and Chen–LeBrun–Weber metrics, starting from the round sphere.
The project was developed with substantial AI assistance, particularly in finding the differential inequalities underlying the compactness and noncollapsing analysis and the curvature identities used in the rigidity and uniqueness arguments. The central strategy, studying moduli spaces of toric conical Einstein metrics and applying a virtual boundary-counting argument, originated entirely with the authors, inspired by gauge-theoretic moduli-space methods, especially compactness and boundary counts in Donaldson theory. The authors also designed the rod structures for the initial existence results and formulated the conjectured diffeomorphism classification. At the authors’ request, building on a mild extension of our strategy and results established earlier in the project, AI then developed the entire proof of the extension to the full classification. The paper includes a detailed account of these contributions.