# Istvan Racz: On construction of Riemannian three-spaces with smooth generalized inverse mean curvature flows

**Time: **
Thu 2019-12-05 10.00 - 11.00

**Location: **
Institut Mittag-Leffler, Seminar Hall Kuskvillan

**Participating: **
Istvan Racz, Wigner Research Center for Physics

### Abtract

Choose a smooth three-dimensional manifold \(\Sigma\) that is smoothly foliated by topological two-spheres, and also a smooth flow on \(\Sigma\) such that the integral curves of it intersect the leaves of the foliation precisely once. Choose also a smooth Riemannian three-metric \(h_{ij}\) on \(\Sigma\) such that the foliating two-spheres are mean convex with respect to it. Then, by altering suitably the lapse and shift of the flow but keeping the two-metrics induced on the leaves of the foliation fixed a large variety of Riemannian three-geometries is constructed on \(\Sigma\) such that the foliation, we started with, gets to be a smooth generalized inverse mean curvature foliation, the prescribed flow turns out to be a generalized inverse mean curvature flow. All this is done such that the scalar curvature of the constructed three-geometries is not required to be non-negative. Furthermore, each of the yielded Riemannian three-spaces are such that the Geroch mass is non-decreasing, and also if the metric \(h_{ij}\) we started with is asymptotically flat then for the constructed three-geometries the positive mass theorem also holds.