# Toroidal trapped surfaces and isoperimetric inequalities

## TL;DR

This paper analytically constructs and analyzes unstable trapped toroidal surfaces within spherically symmetric initial data for Einstein's equations, revealing their existence in regions surrounding Schwarzschild horizons.

## Contribution

It introduces the first analytical and numerical construction of trapped toroidal surfaces in spherically symmetric spacetimes, expanding understanding of trapped surfaces topology.

## Key findings

- Existence of infinite trapped tori in initial data
- Analytical and numerical identification of marginally trapped tori
- Trapped tori are unstable and located near Schwarzschild horizons

## Abstract

We analytically construct an infinite number of trapped toroids in spherically symmetric Cauchy hypersurfaces of the Einstein equations. We focus on initial data which represent "constant density stars" momentarily at rest. There exists an infinite number of constant mean curvature tori, but we also deal with more general configurations. The marginally trapped toroids have been found analytically and numerically; they are unstable. The topologically toroidal trapped surfaces appear in a finite region surrounded by the Schwarzschild horizon.