# Einstein Metrics, 4-Manifolds, and Gravitational Instantons

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A Riemannian metric is said to be **Einstein** if it has constant Ricci curvature. Certain peculiar features of 4-dimensional geometry make dimension four into a “Goldilocks zone” for Einstein metrics, with just the right amount of local ﬂexibility managing to coexist with strong global rigidity results. This talk will ﬁrst describe some aspects of the interplay between Einstein metrics and smooth topology on compact symplectic 4-manifolds without boundary. We will see how ideas from Kähler and conformal geometry allow us to construct Einstein metrics on many such manifolds, while a complimentary tool-box shows that these existence results are optimal in certain specific contexts. The talk will then conclude with a brief discussion of analogous results concerning complete Ricci-ﬂat 4-manifolds.