Naked singularities in initial surfaces

We consider a singular hypersurface {Σ}, carrying time-symmetric initial data for the Einstein equations. We assume that the area of the arbitrary two-sphere, enclosing the singularity, is bounded from below by some positive constant. A conformally flat ``ring,'' or ``pancake'' singularities having sufficiently large Euclidean radius, can serve as examples. We prove that if the Arnowitt-Deser-Misner mass associated with such a hypersurface is small enough, then this singularity is naked (i.e., it is not entirely surrounded by an apparent horizon). We suggest that a similar effect appears also for general (i.e., non-time-symmetric) hypersurfaces.