Curvature Estimates in Asymptotically Flat Manifolds of Positive Scalar Curvature

Suppose that (Mn, g) is an asymptotically flat Riemannian spin manifold of positive scalar curvature. The positive mass theorem [1, 2, 3] states that the total mass of the manifold is always positive, and is zero if and only if the manifold is flat. This result suggests that there should be an inequality which bounds the Riemann tensor in terms of the total mass and implies that curvature must become small when the total mass tends to zero. In [4] such curvature estimates were derived in the context of General Relativity for 3-manifolds being hypersurfaces in a Lorentzian manifold. In the present paper, we study the problem more generally on a Riemannian manifold of dimension n ≥ 3. Our curvature estimates then give a quantitative relation between the local geometry and global properties of the manifold. The main difficulty in higher dimensions is to bound the Weyl tensor (which for n = 3 vanishes identically). Our basic strategy for controlling the Weyl tensor can be understood from the following simple consideration. The existence of a parallel spinor in an open set U ⊂M implies that the manifold is Ricci flat in U . Thus it is reasonable that by getting suitable estimates for the derivatives of a spinor, one can bound all components of the Ricci tensor. This method is used in [4], where a solution of the Dirac equation is analyzed using the Weitzenböck formula. But the local existence of a parallel spinor does not imply that the Weyl tensor vanishes. This is the underlying reason why in dimension n > 3, our estimates cannot be obtained by looking at one spinor, but we must consider a family (ψ)i=1,...,2[n/2] of solutions of the Dirac equation. Out of these solutions we form the so-called spinor operator Px. The curvature tensor can be bounded in terms of suitable derivatives of Px, and an integration-by-parts argument, the Weitzenböck formula, and a-priori estimates for the spinor operator give the desired result. We now give the precise statement of our result. For simplicity, we consider only one asymptotically flat end. The following definition immediately generalizes that used in [4]; for a slightly more general definition see [3].