  This paper focuses on supergravity duals of BPS states in N=4 super
Yang-Mills. In order to describe these duals, we begin with a sequence of
breathing mode reductions of IIB supergravity: first on S^3, then S^3 x S^1,
and finally on S^3 x S^1 x CP^1. We then follow with a complete supersymmetry
analysis, yielding 1/8, 1/4 and 1/2 BPS configurations, respectively (where in
the last step we take the Hopf fibration of S^3). The 1/8 BPS geometries, which
have an S^3 isometry and are time-fibered over a six-dimensional base, are
determined by solving a non-linear equation for the Kahler metric on the base.
Similarly, the 1/4 BPS configurations have an S^3 x S^1 isometry and a
four-dimensional base, whose Kahler metric obeys another non-linear,
Monge-Ampere type equation.
  Despite the non-linearity of the problem, we develop a universal bubbling AdS
description of these geometries by focusing on the boundary conditions which
ensure their regularity. In the 1/8 BPS case, we find that the S^3 cycle
shrinks to zero size on a five-dimensional locus inside the six-dimensional
base. Enforcing regularity of the full solution requires that the interior of a
smooth, generally disconnected five-dimensional surface be removed from the
base. The AdS_5 x S^5 ground state corresponds to excising the interior of an
S^5, while the 1/8 BPS excitations correspond to deformations (including
topology change) of the S^5 and/or the excision of additional droplets from the
base. In the case of 1/4 BPS configurations, by enforcing regularity
conditions, we identify three-dimensional surfaces inside the four-dimensional
base which separate the regions where the S^3 shrinks to zero size from those
where the S^1 shrinks.
