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


Chen-Yun Lin

10:30:00 - 12:00:00

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

The scalar curvature problem arises naturally in general relativity as space-like hypersurface in the underlying space-time. In 1993, Bartnik introduced a quasi-spherical construction of metrics of prescribed scalar curvature on 3-manifolds. This quasi-spherical ansatz has a background foliation with round metrics and converts the problem into a semi-linear parabolic equation. It is also known by work of R. Hamilton and B. Chow that the evolution under the Ricci flow of an arbitrary initial metric $g_0$ on $S^2$, suitably normalized, exists for all time and converges to the round metric.
In this talk, we describe a construction of metrics of prescribed scalar curvature using solutions to the Ricci flow. Considering background foliations given by Ricci flow solutions, we obtain a parabolic equation similar to Bartnik’s. We discuss conditions of scalar curvature 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 3-metrics with outermost minimal surfaces are obtained.

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