Rigidity of marginally outer trapped (hyper)surfaces with negative 𝜎-constant

Abstract

In this paper we generalize the main result of [13] in two different situations: in the first case for MOTSs of genus greater than one and, in the second case, for MOTSs of high dimension with negative σ-constant. In both cases we obtain a splitting result for the ambient manifold when it contains a stable closed MOTS which saturates a lower bound for the area (in dimension 2) or for the volume (in dimension ≥ 3). These results are extensions of [21,Theorem 3] and [20, Theorem 3] to general (non-time-symmetric) initial data sets.

Initial Data Rigidity Results

“…We say that a symmetric (0, 2)-tensor field P is (n − 1)-convex if, at every point, the sum of the smallest (n − 1) eigenvalues of P with respect to g is non-negative. In particular, if P is (n −1)-convex, then tr P ≥ 0 for every hypersurface ⊂ M. This convexity condition has been used by the third-named author in [20] in a related context.…”

Rigidity of free boundary MOTS

“…We say that K is n-convex with respect to g if, at every point of M, the sum of the smallest n eigenvalues of K with respect to g is nonnegative (see [9,15]). In particular, if K is n-convex, then tr S K ≥ 0 for every hypersurface S ⊂ M. We say that Σ is outer volume-minimizing if vol(Σ) ≤ vol(S) for every hypersurface S outside of, and homologous to, Σ. Theorem 5.1.…”

“…Item (4) follows directly from equation (15) and the fact that µ = |J| on each leaf Σ t . Finally, since |(ι ̺ π) ⊤ | = II ∂M (N t , N t ) = X, ν t = 0 along ∂Σ t , ∂Σ t is totally geodesic in (Σ t , γ t ), and Σ t is free boundary in (M, g), we obtain that (ι ̺ π) ⊤ = 0 and II ∂M = 0 along ∂Σ t for each t ∈ [0, ǫ).…”

Rigidity of outermost MOTS: the initial data version

“…The following was a key element in the proofs of Theorems 1.2 and 3.1 in [6] (see also [8,11]). Lemma 2.3.…”