Minimal two-spheres in three-spheres

Abstract

We prove that any manifold diffeomorphic to $S^3$ and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three manifolds. We apply our methods to solve a problem posed by S.T. Yau in 1987 on whether the planar two-spheres are the only minimal spheres in ellipsoids centered about the origin in ${\mathbb{R}}^4$. Finally, considering the example of degenerating ellipsoids we show that the assumptions in the multiplicity one conjecture and the equidistribution of widths conjecture are in a certain sense sharp.

41 pages

References in corpus (2)

• Mean curvature flow with surgery
• On the classification of Heegaard splittings

Cited by in corpus (3)

• Determining a Riemannian Metric from Minimal Areas
• The p-widths of a surface
• On self shrinkers of medium entropy in ${\mathbb{R}}^4$