Minimal surfaces and 3-manifolds of non-negative Ricci curvature

Compact embedded minimal surfaces in $\mathbb{S}^2 \times \mathbb{S}^1$

“…We will begin by describing the intersection of two compact minimal surfaces in S 2 × S 1 (r). The ideas in the proof are adapted from those of Frankel [4] (see also [1,5]). In [19,Theorem 4.3], a similar result is proved for properly embedded minimal surfaces in S 2 × R by using a different approach.…”

“…One of the properties of these 3-manifolds is the fact that they generally admit compact (without boundary) embedded minimal surfaces. Many authors have contributed to the study of compact minimal surfaces in order to understand the geometry of the 3-manifold (see, for instance, [1,5,14,21]). In 1970, Lawson [11] proved that any compact orientable surface can be minimally embedded in the constant sectional curvature 3-sphere S 3 .…”

On minimal hypersurfaces of nonnegatively Ricci curved manifolds

“…Suppose that there exists a smooth minimal immersion of a smooth compact connected d-1 dimensional manifold T without boundary in M. Then In this sense, we point out its relation to the works of Anderson [7] and Anderson and Rodriguez [8].…”

A Splitting Theorem for Scalar Curvature

“…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 place of Ricci curvature and where area-minimizing cylinders stand in for length-minimizing geodesic lines. 1 Theorem 1.1 follows from the work of M. Anderson and L. Rodríguez [2] when we impose the much stronger assumption of bounded, non-negative Ricci curvature. The strategy of M. Anderson and L. Rodríguez [2] has been refined by G. Liu [13] to classify complete, non-compact Riemannian 3-manifolds with non-negative Ricci curvature.…”

“…1 Theorem 1.1 follows from the work of M. Anderson and L. Rodríguez [2] when we impose the much stronger assumption of bounded, non-negative Ricci curvature. The strategy of M. Anderson and L. Rodríguez [2] has been refined by G. Liu [13] to classify complete, non-compact Riemannian 3-manifolds with non-negative Ricci curvature. These ideas have been developed by the first-and second-named authors to establish the following scalar-curvature rigidity result for asymptotically flat 3-manifolds which had been conjectured by R. Schoen.…”