Toroidal trapped surfaces and isoperimetric inequalities

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.

Figure 1

The shapes of MTT’s obtained for $\beta =3+2\sqrt{2},6,10,20$. All these surfaces are contained within the star.

Figure 2

The shapes of MTT’s obtained for $\beta =5$, 4, 2.58, 1.5, 1.1, 1.01.