  The Schwarzschild spacetime metric of negative mass is well-known to contain
a naked singularity. In a spacelike slice, this singularity of the metric is
characterized by the property that nearby surfaces have arbitrarily small area.
We develop a theory of such "zero area singularities" in Riemannian manifolds,
generalizing far beyond the Schwarzschild case (for example, allowing the
singularities to have nontrivial topology). We also define the mass of such
singularities. The main result of this paper is a lower bound on the ADM mass
of an asymptotically flat manifold of nonnegative scalar curvature in terms of
the masses of its singularities, assuming a certain conjecture in conformal
geometry. The proof relies on the Riemannian Penrose Inequality. Equality is
attained in the inequality by the Schwarzschild metric of negative mass. An
immediate corollary is a version of the Positive Mass Theorem that allows for
certain types of incomplete metrics.
