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

TL;DR

This paper extends Bartnik's quasi-spherical construction for prescribed scalar curvature metrics on 3-manifolds by exploring conformal and Ricci flow foliations, providing new existence results and examples of asymptotically flat metrics.

Contribution

It generalizes the background foliation assumptions in Bartnik's construction, establishing conditions for solvability and constructing new examples of asymptotically flat 3-metrics.

Findings

Conditions for solvability of the parabolic equation are established.

Existence of asymptotically flat metrics with prescribed scalar curvature is demonstrated.

New examples with outermost minimal surfaces are constructed.

Abstract

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.