  Eight different refinements of trapped surfaces are proposed, of three basic
types, each intended as potential stability conditions. Minimal trapped
surfaces are strictly minimal with respect to the dual expansion vector. Outer
trapped surfaces have positivity of a certain curvature, related to surface
gravity. Increasingly (future, respectively past) trapped surfaces generate
surfaces which are more trapped in a (future, respectively past) causal
variation, with three types: in any such causal variation, along the expansion
vector, and in some such causal variation. This suggests a definition of doubly
outer trapped surface involving two independent curvatures. This in turn
suggests a definition of involute trapped surface. Adding a weaker condition,
the eight conditions form an interwoven hierarchy, with four independent
relations which assume the null energy condition, and another holding in a
special case of symmetric curvature.

General Relativity and Quantum Cosmology;Differential Geometry