Involute, minimal, outer and increasingly trapped surfaces

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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.