Curvature and Injectivity Radius Estimates for Einstein 4-manifolds

It is of fundamental interest to study the geometric and analytic properties of compact Einstein manifolds and their moduli. In dimension 2 these problems are well understood. A 2-dimensional Einstein manifold, (M, g), has constant curvature, which after normalization, can be taken to be −1, 0 or 1. Thus, (M, g) is the quotient of a space form and the metric, g, is completely determined by the conformal structure. For fixed M, the moduli space of all such g admits a natural compactification, the Deligne-Mumford compactification, which has played a crucial role in geometry and topology in the last two decades, e.g. in establishing Gromov-Witten theory in symplectic and algebraic geometry. In dimension 3, it remains true that Einstein manifolds have constant sectional curvature and hence are quotients of space forms. An essential portion of Thurston’s geometrization program can be viewed as the problem of determining which 3manifolds admit Einstein metrics. The moduli space of Einstein metrics on a 3dimensional manifold is also well understood. As a consequence of Mostow rigidity, the situation is actually simpler than in two-dimensions. In dimension 4 however, the class of Einstein metrics is significantly more general than that of metrics of constant curvature. For example, almost all complex surfaces with definite first Chern class admit Kähler-Einstein metrics. Still, the existence of an Einstein metric does impose strong constraints on the underlying 4-manifold. Hence, it is natural to look for sufficient conditions for a closed 4-manifold to admit an Einstein metric. Any approach to this existence problem by geometric analytic methods, e.g. by Ricci flow, will lead to the question of how, in limiting cases, solutions to the Einstein equation can develop singularities, or equivalently, how Einstein metrics can degenerate. On the other hand, most Einstein 4-manifolds have nontrivial moduli spaces. These moduli spaces and their natural compactifications are differentiable invariants of underlying smooth 4-manifolds. Thus, one wants to understand the geometry of such moduli spaces and their compactifications. Here one can normalize the Einstein constant, λ, to be −3, 0 or 3, and in the (scale invariant) case, λ = 0,