In this paper, we prove that there exists a universal constant C, depending only on positive integers n ≥ 3 and p ≤ n − 1, such that if M n is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball B n+k whose size of the traceless second fundamental form is less than C, then the pth cohomology group of M n vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball B 2+k .
We prove several rigidity results related to the spacetime positive mass theorem. A key step is to show that certain marginally outer trapped surfaces are weakly outermost. As a special case, our results include a rigidity result for Riemannian manifolds with a lower bound on their scalar curvature.
In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by 4π/c, where c > 0 is a lower bound on a natural energy-momentum term. We then consider the rigidity that results for stable, or weakly outermost, marginally outer trapped 2-spheres that achieve this upper bound on the area. In particular, we prove a splitting result for 3-dimensional initial data sets analogous to a result of Bray, Brendle and Neves [10] concerning area minimizing 2-spheres in Riemannian 3-manifolds with positive scalar curvature. We further show that these initial data sets locally embed as spacelike hypersurfaces into the Nariai spacetime. Connections to the Vaidya spacetime and dynamical horizons are also discussed.
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.
Abstract. In this paper we obtain an analogue of Toponogov theorem in dimension 3 for compact manifolds M 3 with nonnegative Ricci curvature and strictly convex boundary ∂M . Here we obtain a sharp upper bound for the length L(∂Σ) of the boundary ∂Σ of a free boundary minimal surface Σ 2 in M 3 in terms of the genus of Σ and the number of connected components of ∂Σ, assuming Σ has index one. After, under a natural hypothesis on the geometry of M along ∂M , we prove that if L(∂Σ) saturates the respective upper bound, then M 3 is isometric to the Euclidean 3-ball and Σ 2 is isometric to the Euclidean disk. In particular, we get a sharp upper bound for the area of Σ, when M 3 is a strictly convex body in R 3 , which is saturated only on the Euclidean 3-balls (by the Euclidean disks). We also consider similar results for free boundary stable CMC surfaces.
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