  Let X be a complex manifold and c a simple closed curve in X. We address the
question: What conditions on c ensure the existence of a 1-dimensional complex
subvariety V with boundary c in X. When X = C^n, an answer to this question
involves the polynomial hull of gamma. When X = P^n, complex projective space,
the projective hull hat{c} of c comes into play. One always has V contained in
hat{c}, and for analytic curves they conjecturally coincide.
  In this paper we establish an approximate analogue of this idea which holds
without the analyticity of c. We characterize points in hat{c} as those which
lie on a sequence of analytic disks whose boundaries converge down to c. This
is in the spirit of work of Poletsky and of Larusson-Sigurdsson, whose work is
essential here.
  The results are applied to construct a remarkable example of a closed curve c
in P^2, which is real analytic at all but one point, and for which the closure
of hat{c} is W \cup L where L is a projective line and W is an analytic
(non-algebraic) subvariety of P^2 - L. Furthermore, hat{c} itself is the union
of W with only two points on L.
