In this paper we modify an argument of Bray, Brendle and Neves to prove an area comparison result (Theorem 2) for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature.This theorem is a variant of a comparison theorem (Theorem 3.2 (d) in the 1978 paper) of Heintze and Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient manifold are weaker, but the assumptions on the surface are considerably more restrictive.We then use our comparison theorem to provide a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature that were first proved

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.

We dedicate this paper to Gregory Galloway on the occasion of his 70 th birthday. AbstractWe show that a Riemannian 3-manifold with nonnegative scalar curvature is flat if it contains an area-minimizing cylinder. This scalar-curvature analogue of the classical splitting theorem of J. Cheeger and D. Gromoll (1971) was conjectured by D. Fischer-Colbrie and R. Schoen (1980) and by M. Cai and G. Galloway (2000).Let .M; g/ be a connected, orientable, complete Riemannian 3-manifold with nonnegative scalar curvature. D. Fischer-Colbrie and R. Schoen show in [10] that a connected, orientable, complete stable minimal immersion into .M; g/ is conformal to a plane, a sphere, a torus, or a cylinder. They conjecture that .M; g/ is flat if the immersion is conformal to the cylinder; cf. remark 4 in [10]. M. Cai and G. Galloway point out a counterexample obtained from flattening standard R S 2 near R fgreat circleg in their concluding remark in [7]. They ask if the conjecture holds under the additional assumption that the immersion be "suitably" area-minimizing. In this paper, we prove the following result: THEOREM 1.1. Let .M; g/ be a connected, orientable, complete Riemannian 3manifold with nonnegative scalar curvature. Assume that .M; g/ contains a properly embedded surface S M that is both homeomorphic to the cylinder and absolutely area-minimizing. Then .M; g/ is flat. In fact, a cover of .M; g/ is isometric to standard S 1 R 2 upon scaling.Note that this result is in satisfying analogy with the classical splitting theorem of J. Cheeger and D. Gromoll [9] in dimension 3, where scalar curvature takes the

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