This volume examines elliptic PDEs (partial differential equations) on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower

Ricci curvature bounds, specifically establishing continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schr|dinger operators, generalized Yamabe constants, and eigenvalues of the Hodge Laplacian, with respect to the Gromov-Hausdorff topology.

The nonvanishing components of the Maxwell-f(R) field equations, (5) and (6), when f(R) = R+[alpha][R.sup.2] take the form (the detailed calculations of the

Ricci curvature tensor are given in Appendix B)

where [DELTA] is the Laplacian associated with the connection D in the normal bundle of [phi](S), [??] stands for the Simons operator [13], Ric is the

Ricci curvature of [bar.M], and N denotes the unit normal vector field of [phi].

where k is the Einstein constant, [T.sub.ab] is the energy-momentum, and [R.sub.ab] is the

Ricci curvature tensor which represents geometry of the spacetime in presence of energy-momentum.

Next we make use of an exact result valid for a 3 + 1 decomposition of space-time: [G.sub.tt] = R/2, where R is the scalar

Ricci curvature of the three-dimensional hypersurface [40].

for any X = ([X.sub.1], [X.sub.2]), Y = ([Y.sub.1], [Y.sub.2]) [member of] T([M.sub.1] x [M.sub.2]), where S, [S.sub.1] and [S.sub.2] are respectively the

Ricci curvature tensors of the Riemannian manifolds ([M.sub.i] x [M.sub.2], g), ([M.sub.1], [g.sub.1]) and ([M.sub.2], [g.sub.2]).

(1) If the transversal

Ricci curvature of F is positive-definite, then any [L.sup.2]-basic harmonic 1-forms [phi] with [phi] [member of] [S.sub.B] are trivial.

Key words:

Ricci curvature tensors, Einstein Field Equations, Black hole, Vacuum Solutions

where R and S are respectively Riemann and

Ricci curvature tensors.

On the other hand, if one chooses to sample the manifold using

Ricci curvature, as proposed in [41], [43], one needs to encode just one additional, number, namely the

Ricci curvature Ric (or, rather, its absolute value).