Mean curvature flow and Heegaard surfaces in lens spaces
Abstract
We prove that the moduli space of mean convex two-spheres embedded in complete, orientable $3$-dimensional Riemannian manifolds with nonnegative Ricci curvature is path-connected. This result is sharp in the sense that neither of the conditions of (strict) mean convexity, completeness, and nonnegativity of the Ricci curvature can be dropped or weakened. We also study the number of path components of mean convex Heegaard tori, again in ambient manifolds with nonnegative Ricci curvature. We prove that there are always either one or two path components and this number does not only depend on the homotopy type of the ambient manifold. We give a precise characterisation of the two cases and also discuss what happens if the mean convexity condition is weakened to nonnegative mean curvature.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Geometry and complex manifolds
Therapeutic Uses of Natural Elements
Publication details
- Journal / source
- Communications in Analysis and Geometry
- Publisher
- International Press of Boston
- Volume
- 34
- Issue
- 3
- Pages
- 663–693
- Publication date
- 1 January 2026
- Language
- EN