Estimates for the Riemann Curvature Tensor

Abstract

This chapter is devoted to the proof of Theorem M7 in terms of the fundamental quantities Q, $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Q}$. These quantities can be expressed, according to (3.5.1), as weighted integrals of the null components of $\hat L_0 R,\hat L_0 R,\hat L^2 _0 R,\hat L_0 \hat L_T R and \hat L_S \hat L_T R$along the null hypersurfaces C(λ) and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{C}$(ν). We recall their explicit expressions:

$$\begin{gathered} Q(\lambda ,\nu ) = Q_1 (\lambda ,\nu ) + Q_2 (\lambda ,\nu ) \hfill \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Q} (\lambda ,\nu ) = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Q} _1 (\lambda ,\nu ) + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Q} _2 (\lambda ,\nu ) \hfill \\ \end{gathered}$$