m at h . D G ] 2 1 A pr 1 99 8 LEAST AREA TORI AND 3 - MANIFOLDS OF NONNEGATIVE SCALAR CURVATURE : THE C ∞ CASE
نویسنده
چکیده
The following conjecture arises from remarks in Fischer-Colbrie-Schoen ([FCS], Remark 4, p. 207): If (M, g) is a complete Riemannian 3-manifold with nonnegative scalar curvature and if Σ is a two-sided torus in M which is suitably of least area then M is flat. Such a result, as Fischer-Colbrie and Schoen commented, would be an interesting analogue of the Cheeger-Gromoll splitting theorem. Here we present a proof of this conjecture assuming Σ is of least area in its isotopy class. We also present some examples which show that it is not sufficient to assume Σ is stable. Our version of the conjecture is a consequence of the following local result, which is the main result of the paper.
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