Seminar

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

#### General Relativity, Geometry and Analysis: beyond the first 100 years after Einstein

#### 05 December 10:00 - 11:00

#### Istvan Racz - Wigner Research Center for Physics

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.

Organizers

Lars Andersson

Max Planck Institute for Gravitational Physics (Albert Einstein Institute)

Mattias Dahl

KTH Royal Institute of Technology

Philippe G. LeFloch

Sorbonne University

Richard Schoen

University of California, Irvine