On Area Comparison and Rigidity Involving the Scalar Curvature

Abstract

We prove a splitting theorem for Riemannian n-manifolds with scalar curvature bounded below by a negative constant and containing certain area-minimising hypersurfaces Σ (Theorem 3). Thus we generalise [25, Theorem 3] by Nunes. This splitting result follows from an area comparison theorem for hypersurfaces with non-positive σ-constant (Theorem 4) that generalises [23, Theorem 2]. Finally, we will address the optimality of these comparison and splitting results by explicitly constructing several examples.

Free Boundary Hypersurfaces with Non-positive Yamabe Invariant in Mean Convex Manifolds

“…In higher dimensions, Cai [9] showed a local splitting of an n-dimensional manifold M with nonegative escalar curvature containing a volume-minimizing hypersurface that does not admit a metric of positive scalar curvature. In this direction, Moraru [21] proved a natural extension of the rigidity result contained in [23].…”

“…This, together with the resolution of the Yamabe problem for compact manifolds with boundary, implies that each hypersurface has the same volume. For this volume comparison, we adapt a technique developed by Moraru [21]. After, we exhibited an isometry from (−ε, ε) × Σ into a neighborhood of Σ.…”

“…When Q 0,1 g (Σ, ∂Σ) is finite and Q 0,1 g (Σ, ∂Σ) < Q 0,1 g (B n−1 , ∂B n−1 ), there exists a smooth metric of flat scalar curvature and mean curvature on the boundary equal to K that has the same sign as Q 0,1 g (Σ, ∂Σ). There are a lot of interesting papers related with this subject, we indicate for instance [13], [14], [15], [19], [21] and [1].…”

“…Let u t be a positive function on Σ t satisfying g t = u 4 n−3 t g t . Now, we will adapt the method introduced in [21] to establish the volume comparison. First multiplying (4.6) by u 2 t and integrating along Σ t it becomes…”

Metrics of positive scalar curvature and unbounded min-max widths

“…Since the proof of the positive mass conjecture in general relativity by Schoen and Yau [23], and Witten [26], the rigidity phenomena involving the scalar curvature has been fascinating the geometers. These results play an important role in modern differential geometry and there is a vast literature about it, see ( [1,3,4,5,6,7,8,12,15,16,17,18,19,22]). Many of these works concern rigidity phenomena involving the scalar curvature and the area of minimal surfaces of some kind in three-manifolds.…”