Rigidez de MOTS com fronteira livre em um conjunto de dados iniciais

Abstract

In this work, we present some rigidity results for free boundary MOTS in initial data sets that satisfy certain natural geometric and energy conditions. The first result deals with compact free boundary hypersurfaces Σ in Riemannian manifolds (Mn+1, g) with boundary, where the scalar curvature of M satisfies RM ≥ −(n + 1)n or RM ≥ 0 and the mean curvature of Σ satisfies HΣ ≤ n or HΣ ≤ 0, respectively. In the proof, we use the theory of free boundary MOTS in initial data sets with boundary. This result was motivated by the work of G. J. Galloway and H. C. Jang, who obtained the same result for manifolds M with compact boundary Σ. In the second result, a lower bound is presented for the volume of compact stable free boundary MOTS Σ in initial data sets (Mn+1, g, K), with n ≥ 3, assuming that Σ has negative Yamabe invariant. In the case where the equality holds, it is possible to prove that an outer neighborhood of Σ in M is isometric to ([0, t)×Σ, dt2+gΣ). This result was motivated by the work of A. Mendes, who studied the case of closed MOTS, and the work of A. Barros and C. Cruz, who obtained sharp estimates for the volume of compact stable minimal hypersurfaces with nonpositive Yamabe invariant satisfying the free boundary condition in a Riemannian manifold with lower bounds on the scalar curvature and the mean curvature of the boundary, and the rigidity result when the volume saturates the inequality.