Minimal 2-spheres in 3-spheres

Abstract

We prove that any manifold diffeomorphic to ${\mathbb{S}}^3$ and endowed with a generic metric contains at least two embedded minimal $2$-spheres. The existence of at least one minimal $2$-sphere was obtained by Simon and 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 $3$-manifolds. We apply our methods to solve a problem posed by S. T. Yau in 1987 on whether the planar $2$-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 $1$ conjecture and the equidistribution of widths conjecture are in a certain sense sharp.