On a Liu--Yau type inequality for surfaces

TL;DR

This paper establishes a Liu--Yau type inequality relating the mean curvature vector of a surface in a spacetime to an isometric immersion into Euclidean space, with implications for rigidity and isometric immersions in Minkowski spacetime.

Contribution

It extends Liu--Yau inequalities to more general settings, including higher dimensions and weaker conditions, and proves rigidity results for isometric immersions in Minkowski space.

Findings

Proves an inequality relating mean curvature vectors and isometric immersions.

Characterizes conditions for equality leading to local isometric immersions in Minkowski space.

Establishes a stronger rigidity theorem in higher-dimensional Minkowski space.

Abstract

Let $\Omega$ be a compact and mean-convex domain with smooth boundary $\Sigma:=\partial\Omega$, in an initial data set $(M^3,g,K)$, which has no apparent horizon in its interior. If $\Sigma$ is spacelike in a spacetime $(\E^4,g\_\E)$ with spacelike mean curvature vector $\mathcal{H}$ such that $\Sigma$ admits an isometric and isospin immersion into $\mathbb{R}^3$ with mean curvature $H\_0$, then: \begin{eqnarray*} \int\_{\Sigma}|\mathcal{H}|d\Sigma\leq\int\_{\Sigma}\frac{H\_0^2}{|\mathcal{H}|}d\Sigma. \end{eqnarray*} If equality occurs, we prove that there exists a local isometric immersion of $\Omega$ in $\mathbb{R}^{3,1}$ (the Minkowski spacetime) with second fundamental form given by $K$. In Theorem liu-yau-minkowski, we also examine, under weaker conditions, the case where the spacetime is the $(n+2)$-dimensional Minkowski space $\mathbb{R}^{n+1,1}$ and establish a stronger rigidity result.