Minimal spheres and scalar curvature

Abstract

In 1982, S.-T. Yau conjectured that there exist four distinct embedded minimal two-spheres in any manifold diffeomorphic to $S^3$. Wang-Zhou confirmed this conjecture for Riemannian three-spheres when the metric is bumpy or has positive Ricci curvature. We prove the following quantitative version of their theorem. Suppose that $(S^3,g)$ has positive Ricci curvature and scalar curvature $R_g\ge \Lambda_0>0$. Then there exist four distinct embedded minimal two-spheres $\Sigma_1,\ldots,\Sigma_4\subset (S^3,g)$ such that $\operatorname{area}_{g}(\Sigma_i)\le 12\pi(i+1)/\Lambda_0$ for every $i=1,\ldots,4$. We apply this result to a problem posed by S.-T. Yau in 1987 on whether the planar two-spheres are the only minimal spheres in ellipsoids centered at the origin in $\mathbb R^4$. Haslhofer-Ketover proved that ellipsoids with one sufficiently large semi-axis contain at least one non-planar embedded minimal two-sphere. We prove that such ellipsoids contain at least three non-planar embedded minimal two-spheres.