The moduli space of mean convex two-spheres in complete orientable 3-manifolds with nonnegative Ricci curvature is path-connected; mean convex Heegaard tori have one or two path components depending on a precise characterisation.
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Mean Curvature Flow and Heegaard Surfaces in Lens Spaces
The moduli space of mean convex two-spheres in complete orientable 3-manifolds with nonnegative Ricci curvature is path-connected; mean convex Heegaard tori have one or two path components depending on a precise characterisation.