Einstein Metrics and Metrics with Bounds on Ricci Curvature

Abstract

There is a well-developed theory of the behavior of Riemannian metrics on smooth manifolds, which have uniform bounds on the sectional curvature K. The compactness theorem of Cheeger-Gromov [Ch], [Gr] implies that the space of metrics satisfying the bounds

$$\left| K \right| \leqslant \Lambda ,{\text{vol}} \geqslant v,{\text{diam}} \leqslant D$$

(0.1)

is C^{1, α} precompact. Thus, given any sequence of metrics g _i satisfying the bounds (0.1), there is a subsequence {i′} and a sequence of diffeomorphisms φ_L′ of M, such that the isometric metrics g′ _p = (φ_i′)*g_i′ converge, in the C^{1, α′} topology on M, to a C^{1, α} metric g_∞ on M, ∀α′ < α < 1. If the volume or diameter bounds are removed in (0.1), one no longer has such compactness, but the degeneration of the sequence {g _i } is well understood, through the workds of Cheeger-Gromov [CG₁], [CG₂] and Fukaya [F], cf. also [CGF]. The manifolds (M, g _i ) divide into two regions, the thick part M^ε and the thin part M_ε. Roughly speaking, on M^ε the metrics converge, as above, to a limit C^1,α metric, while the complement L_ε is ε-collapsed along a well-defined topological structure, called an F-structurem or more generally an N-structure.