# Space-time convex functions and sectional curvature

## TL;DR

This paper explores the relationship between sectional curvature bounds and convex functions in Lorentzian manifolds, providing new constructions and applications to trapped submanifolds in space-time geometry.

## Contribution

It establishes a connection between curvature bounds and convex functions, offering a novel construction method and extending results on trapped submanifolds.

## Key findings

- Sectional curvature bounds relate to space-time convex functions.
- A new method to construct such convex functions is introduced.
- Applications include ruling out trapped submanifolds in Lorentzian geometry.

## Abstract

We show that in Lorentzian manifolds, sectional curvature bounds of the form $\mathcal{R}\le K\,$, as defined by Andersson and Howard, are closely tied to space-time convex and $\lambda$-convex ($\lambda>0$) functions, as defined by Gibbons and Ishibashi. Among the consequences are a natural construction of such functions, and an analogue, that applies to domains of a new type, of a theorem of Al\'ias, Bessa and deLira ruling out trapped submanifolds.