Least area tori and 3-manifolds of nonnegative scalar curvature (Q5961422)

This paper is motivated by the following conjecture: Let \(M\) be an orientable 3-manifold of nonnegative scalar curvature. Suppose that \(\Sigma\) is a torus in \(M\) of least area among all nearby surfaces isotopic to it. Then \(M\) splits, i.e., there is a neighbourhood \(U\) of \(\Sigma\) which is isometric to \((-\varepsilon,\varepsilon)\times\Sigma\) and hence is flat. By means of a detailed tensor analysis the authors prove that if \(M\) is analytic, then the conjecture is true. In the global version, it is shown that if \(M\) is geodesically complete, it is isometric to, or doubly covered by, either a flat torus or \(\mathbb{R}\times\Sigma\). If \(M\) is compact with mean convex boundary, it is isometric to, or doubly covered by \([0,\ell] \times \Sigma\).