Extension of symmetries on Einstein manifolds with boundary

TL;DR

This paper studies whether symmetries (Killing fields) on the boundary of Einstein manifolds can extend into the interior, establishing conditions under which such extensions are guaranteed.

Contribution

It proves that boundary symmetries extend into the bulk Einstein metric under mild fundamental group conditions and mean curvature preservation.

Findings

Boundary Killing fields extend to the interior under certain conditions.

Extension holds when the boundary Killing field preserves mean curvature.

Results depend on a mild fundamental group assumption.

Abstract

We investigate the validity of the isometry extension property for (Riemannian) Einstein metrics on manifolds with boundary. Given a metric on the boundary, this is the issue of whether any Killing field of the boundary metric extends to a Killing field of any bulk or filling Einstein metric inducing the given data on the boundary. Under a mild condition on the fundamental group, this is proved to be the case at least when the Killing field preserves the mean curvature of the boundary.