  Regularity and topology conditions are imposed on the exact Type IIB
solutions on $AdS_4 \times S^2 \times S^2 \times \Sigma $ with 16
supersymmetries, which were derived in a companion paper. We construct an
infinite class of regular solutions with varying dilaton, and non-zero 3-form
fluxes. Our solutions may be viewed as the fully back-reacted geometries of
$AdS_5 \times S^5$ (or more generally, Janus) doped with D5 and/or NS5 branes.
The solutions are parametrized by the choice of an arbitrary genus $g$
hyper-elliptic Riemann surface $\Sigma $ with boundary, all of whose branch
points are restricted to lie on a line. For genus 0, the Janus solution with 16
supersymmetries and 6 real parameters is recovered; its topology coincides with
that of $AdS_5 \times S^5$. The genus $g\geq 1$ solutions are parametrized by a
total of $4g+6$ real numbers, $2g-1$ of which are the real moduli of $\Sigma$.
The solutions have $2g+2$ asymptotic $AdS_5 \times S^5$ regions, $g$
three-spheres with RR 3-form charge, and another $g$ with NSNS 3-form charge.
Collapse of consecutive branch points of $\Sigma $ yields singularities which
correspond to D5 and NS5 branes in the probe limit. It is argued that the
AdS/CFT dual gauge theory to each of our solutions consists of a
2+1-dimensional planar interface on which terminate $2g+2$ half-Minkowski
3+1-dimensional space-time $\mathcal{N}=4$ super-Yang-Mills theories.
Generally, the $\\mathcal{N}=4$ theory in each Minkowski half-space-time may
have an independent value of the gauge coupling, and the interface may support
various operators, whose interface couplings are further free parameters of the
dual gauge theory.
