On effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes

TL;DR

The paper analyzes how the null energy condition constrains totally umbilic hypersurfaces in static warped-product spacetimes of Class $\mathfrak{H}$, addressing spacelike STCMC surfaces via Brendle-type Alexandrov rigidity and extending results through generalized Kruskal--Szekeres coordinates to horizons. In the timelike sector, it proves a photon-surface classification with constant umbilicity factor under a Ricci-eigenvalue condition, drawing parallels and contrasts with spherical symmetry results. A key thread is reformulating NEC as a differential inequality for the warp-function perturbation $x=h_T-h$, yielding explicit ODEs whose solutions (e.g., hyperboloids with $h_T=h+Cs^2$) govern rigidity and extension across Killing horizons. The results unify and extend previous spacetime Alexandrov-type theorems to warped-product initial data and provide sharp rigidity statements for STCMC surfaces in hyperboloids, with concrete implications in Schwarzschild-type spacetimes. The work also clarifies how curvature and energy conditions interact to constrain the geometry of photon surfaces, including nuanced behavior inside horizons and in non-spherically symmetric settings.

Abstract

We study the effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes, both in the spacelike and the timelike case, respectively. In the spacelike case, we study totally umbilic warped product graphs and give a full characterization of embedded surfaces with constant spacetime mean curvature using an Alexandrov Theorem by Brendle and Borghini--Fogagnolo--Pinamonti. In the timelike case, we achieve a characterization of photon surfaces with constant umbilicity factor similar to a result by Cederbaum--Galloway.

Theorems & Definitions (33)

• Theorem 4.8
• Corollary 4.9
• Theorem 5.1
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• proof
• Remark 3.2
• Corollary 3.3
• proof