In this dissertation, we study the geometry of null cones in smooth Einstein vacuum spacetimes. We provide the Linfinity estimate for the trace of the null second fundamental form, as well as other geometric quantities such as the shear and torsion tensors, only in term of the L2 curvature flux.; In [12], [14] and [15], the authors have established similar estimates on truncated null hypersurfaces. The method in this dissertation is based on the above three papers, however, with particular attention to the portion near the vertices of null cones, which requires us to prove weighted version of the main estimates in [12], [14] and [15]. In addition, we provide the proofs of the boundedness of 0-order Hodge operators in Besov spaces. Such estimates were stated and used in [12] without a proof. We present a modified version of these estimates. Modification adds complexity to error estimates in Chapter 7.
展开▼
