Abstract
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). © 2018 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc.
Highlights
Let (M, g) be a connected, orientable, complete Riemannian 3-manifold with non-negative scalar curvature
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
Galloway point out a counterexample obtained from flattening standard R × S2 near R × {great circle} in their concluding remark in [7]. They ask if the conjecture holds under the additional assumption that the immersion be “suitably” area-minimizing
Summary
Let (M, g) be a connected, orientable, complete Riemannian 3-manifold with non-negative scalar curvature. Assume that (M, g) contains a properly embedded surface S ⊂ M that is both homeomorphic to the cylinder and absolutely area-minimizing.
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