On a Liu–Yau type inequality for surfaces

Oussama Hijazi, Sebastián Montiel and Simon Raulot

Vol. 272 (2014), No. 1, 177–199

DOI: 10.2140/pjm.2014.272.177

Abstract

Let $\Omega$ be a compact mean-convex domain with smooth boundary $\Sigma :=\partial \Omega$, in an initial data set $\left(M^3,g,K\right)$, which has no apparent horizon in its interior. If $\Sigma$ is spacelike in a spacetime $\left({\mathcal{ℰ}}^4,g_{\mathcal{ℰ}}\right)$ with spacelike mean curvature vector $\mathcal{\mathscr{H}}$ such that $\Sigma$ admits an isometric and isospin immersion into ${ℝ}^3$ with mean curvature $H_0$, then

$${\int \; <!--nolimits-->}_{\Sigma }\left|\mathcal{\mathscr{H}}\right|d\Sigma \le {\int \; <!--nolimits-->}_{\Sigma }\frac{H_0^2}{\left|\mathcal{\mathscr{H}}\right|}d\Sigma .$$

If equality occurs, we prove that there exists a local isometric immersion of $\Omega$ in ${ℝ}^{3,1}$ (the Minkowski spacetime) with second fundamental form given by $K$. We also examine, under weaker conditions, the case where the spacetime is the $\left(n+2\right)$-dimensional Minkowski space ${ℝ}^{n+1,1}$ and establish a stronger rigidity result.

Keywords

manifolds with boundary, Dirac operator, Einstein equations, initial data set, mean curvature, holographic principle

Mathematical Subject Classification 2010

Primary: 53C27, 53C40, 53C80

Milestones

Received: 30 June 2013

Revised: 17 September 2013

Accepted: 1 October 2013

Published: 9 October 2014