A Splitting Theorem for Scalar Curvature

Abstract

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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