Rigidity of min-max minimal spheres in three-manifolds

Abstract

In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or equal to 6 and is not round must have an embedded minimal sphere of area strictly smaller than $4\pi$ and index at most one. If the Ricci curvature is positive we also prove sharp estimates for the width.

References in corpus (2)

• Scalar Curvature Rigidity for asymptotically locally hyperbolic manifolds
• Deformations of the hemisphere that increase scalar curvature

Cited by in corpus (11)

• Minimal two-spheres in three-spheres
• A proof of the multiplicity one conjecture for min-max minimal surfaces in arbitrary codimension
• Classifying sufficiently connected PSC manifolds in $4$ and $5$ dimensions
• On compact Riemannian manifolds with convex boundary and Ricci curvature bounded from below
• Geometric convergence results for closed minimal surfaces via bubbling analysis
• A splitting theorem for scalar curvature
• Rigidity of Free Boundary Surfaces in Compact 3-Manifolds with Strictly Convex Boundary
• Rigidity of volume-minimizing hypersurfaces in Riemannian 5-manifolds
• Appearance of stable minimal spheres along the Ricci flow in positive scalar curvature
• Stability of Convex Spheres
• Equivariant min-max hypersurface in $G$-manifolds with positive Ricci curvature