Curvature estimates in asymptotically flat manifolds of positive scalar curvature

Abstract

AbstractWe consider an asymptotically flat Riemannian spin manifold of positive scalarcurvature. An inequality is derived which bounds the Riemann tensor in terms of thetotal mass and quantifies in which sense curvature must become small when the totalmass tends to zero. 1 Introduction Suppose that (M n ,g) is an asymptotically flat Riemannian spin manifold of positive scalarcurvature. The positive mass theorem [1, 2, 3] states that the total mass of the manifoldis always positive, and is zero if and only if the manifold is flat. This result suggests thatthere should be an inequality which bounds the Riemann tensor in terms of the total massand 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-manifoldsbeing hypersurfaces in a Lorentzian manifold. In the present paper, we study the problemmore generally on a Riemannian manifold of dimension n≥ 3. Our curvature estimatesthen give a quantitative relation between the local geometry and global properties of themanifold.The main difficulty in higher dimensions is to bound the Weyl tensor (which for n= 3vanishes identically). Our basic strategy for controlling the Weyl tensor can be understoodfrom the following simple consideration. The existence of a parallel spinor in an open setU⊂ Mimplies that the manifold is Ricci flat in U. Thus it is reasonable that by gettingsuitable estimates for the derivatives of a spinor, one can bound all components of theRicci tensor. This method is used in [4], where a solution of the Dirac equation is analyzedusing the Weitzenbo¨ck formula. But the local existence of a parallel spinor does not implythat 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