Parabolic constructions of asymptotically flat 3-metrics of prescribed scalar curvature

Abstract

In 1993, Bartnik (J. Differ. Geom. 37(1):37–71) 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 (Shi and Tam in J. Differ. Geom. 62(1):79–125, 2002; Smith in Gen. Relat. Gravit. 41(5):1013–1024, 2009; Smith and Weinstein in Commun. Anal. Geom. 12(3):511–551, 2004) 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.