  We solve the nonlinear evolution of pressureless, irrotational density
fluctuations in a perturbed Robertson-Walker spacetime using a new Lagrangian
method based on the velocity gradient and gravity gradient tensors. Borrowing
results from general relativity, we obtain a set of Newtonian ordinary
differential equations for these quantities following a given mass element.
Using these Lagrangian fluid equations we prove the following results: (1) The
spherical tophat perturbation, having zero shear, is the slowest configuration
to collapse for a given initial density and growth rate. (2) Initial density
maxima are not generally the sites where collapse first occurs. (3) Initially
underdense regions may undergo collapse if the shear is not too small. If the
magnetic part of the Weyl tensor vanishes, the nonlinear evolution is described
purely locally by our equations; this condition holds for spherical,
cylindrical, and planar perturbations and may be a good approximation in other
circumstances. Assuming the vanishing of the magnetic part of the Weyl tensor,
we compute the exact nonlinear gravitational evolution of cold matter. We find
that 56\% of initially underdense regions collapse in an Einstein-de Sitter
universe for a homogeneous and isotropic random field. We also show that, given
this assumption, the final stage of collapse is generically two-dimensional,
leading to strongly prolate filaments rather than Zel'dovich pancakes. While
this result may explain the prevalence of filamentary collapses in N-body
simulations, it is not true in general, suggesting that the magnetic part of
the Weyl tensor need not vanish in the Newtonian limit.
