Killing Initial Data on Totally Umbilical & Compact Hypersurfaces

TL;DR

This paper characterizes compact, totally umbilical hypersurfaces with non-trivial static Killing initial data, showing they are either standard spheres or specific warped products with constant mean curvature.

Contribution

It provides a geometric classification of such hypersurfaces, linking their properties to harmonic curvature and positive scalar curvature, extending understanding of static initial data.

Findings

Hypersurfaces are either standard spheres or warped products with Einstein manifolds.

They have constant mean curvature and harmonic curvature.

Scalar curvature is strictly positive and constant.

Abstract

In this note, we give a geometric characterization of the compact and totally umbilical hypersurfaces that carry a non trivial locally static Killing Initial Data (KID). More precisely, such compact hypersurfaces have constant mean curvature and are isometric to one of the following manifolds: (i) Sn the standard sphere, (ii) a finite quotient of a warped product of a circle with a compact Einstein manifold of positive scalar curvature. In particular, these hypersurfaces have harmonic curvature and strictly positive constant scalar curvature.