Christopher Nerz: Canonical foliations of constant mean curvature

In 1996, Huisken-Yau proved that any three-dimensional asymptotically Schwarzschildean manifold is canonically foliated by closed surfaces of constant mean curvature (CMC). Since then their result has been generalized, and CMC-foliations have proven to be a useful tool for studying asymptotically Euclidean manifolds. These manifolds are used in General Relativity as inital data sets of isolated gravitating systems. Twelve years ago, Rigger proved that an important second class of manifolds used in General Relativity possesses a canonical CMC-foliation: asymptotically hyperbolic manifolds.

In the first part of the talk, I will summarize previous and new existence and uniqueness results for CMC-foliations in both settings, i.e. for asymptotically Euclidean and asymptotically hyperbolic manifolds. I will then explain how the geometric structure given by the CMC-foliations characterizes the asymptotic behaviour of the foliated manifold and important physical quantities. As a last step, I will explain how the CMC-foliations can be used to compare asymptotically hyperbolic and asymptotically Euclidean manifolds (near infinity).