  Up to a conjecture in Riemannian geometry, we significantly strengthen a
recent theorem of Eardley by proving that a compact region in an initial data
surface that is collapsing sufficiently fast in comparison to its
surface-to-volume ratio must contain a future trapped region. In addition to
establishing this stronger result, the geometrical argument used does not
require any asymptotic or energy conditions on the initial data. It follows
that if such a region can be found in an asymptotically flat Cauchy surface of
a spacetime satisfying the null-convergence condition, the spacetime must
contain a black hole with the future trapped region therein. Further, up to
another conjecture, we prove a strengthened version of our theorem by arguing
that if a certain function (defined on the collection of compact subsets of the
initial data surface that are themselves three-dimensional manifolds with
boundary) is not strictly positive, then the initial data surface must contain
a future trapped region. As a byproduct of this work, we offer a slightly
generalized notion of a future trapped region as well as a new proof that
future trapped regions lie within the black hole region.
