Free Boundary Hypersurfaces with Non-positive Yamabe Invariant in Mean Convex Manifolds
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Free Boundary Hypersurfaces with Non-positive Yamabe Invariant in Mean Convex Manifolds A. Barros1
· C. Cruz2
Received: 4 July 2018 © Mathematica Josephina, Inc. 2019
Abstract We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface with non-positive Yamabe invariant in a Riemannian nmanifold with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that is locally volume-minimizing in a manifold M n with scalar curvature bounded from below by a non-positive constant and mean convex boundary, we conclude that locally M splits along . In the case that the scalar curvature of M is at least −n(n − 1) and locally minimizes a certain functional inspired by works of Yau [35] and Andersson-Galloway [4], a neighbourhood of in M is isometric to ((−ε, ε) × , dt 2 + e2t g), where g is Ricci flat with totally geodesic boundary. Keywords Scalar curvature · Stability · Yamabe invariant · Free boundary hypersurfaces · Rigidity · CMC foliations Mathematics Subject Classification Primary 53C42 · 53C21; Secondary 58J60
1 Introduction and Main Results In recent years, rigidity involving the scalar curvature has been studied because these problems are motivated by general relativity and have strong connections with the theory of minimal surfaces, see Reference [9]. Moreover, the existence of an area-
Authors partially supported by CNPq-Brazil.
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A. Barros [email protected] C. Cruz [email protected] http://www.mat.ufc.br
1
Departamento de Matemática-UFC, Fortaleza, CE 60455-760, Brazil
2
Instituto de Matemática-UFAL, Maceió, AL 57072-970, Brazil
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A. Barros, C. Cruz
minimizing surface of some kind, enables us to deduce several rigidity theorems. A deeper result due to Schoen and Yau [32] asserts that any area-minimizing surface in a three-manifold (M, g) with positive scalar curvature is homeomorphic either to S2 or RP2 . Motivated by this, the rigidity of area-minimizing projective planes was studied by Bray et al. [10], while the case of area-minimizing two-spheres was obtained by Bray, Brendle and Neves in Reference [11]. It was also observed by Cai and Galloway [13] that a three-manifold with non-negative scalar curvature is flat in a neighbourhood of a two-sided embedded two-torus which is locally area-minimizing. Given a surface of genus g() > 1, Nunes [27, Theorem 3] has obtained an interesting rigidity result for minimal hyperbolic surfaces in three-manifolds with scalar curvature bounded by a negative constant, whose bounded can be chosen at least −2, then it was shown that the area of this surface is greater than or equal to 4π(g()−1). Furthermore, if equality is attained, then the ambient manifold is locally isometric to the Riemannian product × (−ε, ε), for some ε > 0. There is also an unified point of view with alternative proofs about these cases considered by Micallef and Moraru [24]. For a good reference about other rigidity theorems we refer the reader to Reference [8]. In higher dimensions,
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