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Parabolic Constructions of Asymptotically Flat 3-metrics of Prescribed Scalar Curvature

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.