  In 1993, Bartnik introduced a quasi-spherical construction of metrics of
prescribed scalar curvature on 3-manifolds. Under quasi-spherical ansatz, the
problem is converted into the initial value problem for a semi-linear parabolic
equation of the lapse function. The original ansatz of Bartnik started with a
background foliation with round metrics on the 2-sphere leaves. This has been
generalized by several authors under various assumptions on the background
foliation. In this article, we consider background foliations given by
conformal round metrics, and by the Ricci flow on 2-spheres. We discuss
conditions on the scalar curvature function and on the foliation that guarantee
the solvability of the parabolic equation, and thus the existence of
asymptotically flat 3-metrics with a prescribed inner boundary. In particular,
many examples of asymptotically flat-scalar flat 3-metrics with outermost
minimal surfaces are obtained.
