Minimal spheres and scalar curvature

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 $R_g\ge {\mathrm{\Lambda }}_0>0$. Then there exist four distinct embedded minimal two-spheres ${\mathrm{\Sigma }}_1,\dots ,{\mathrm{\Sigma }}_4\subset (S^3,g)$ such that ${\mathrm{area}}_g({\mathrm{\Sigma }}_i)\le 12\pi (i+1)\mathrm{/}{\mathrm{\Lambda }}_0$ for every $i=1,\dots ,4$. 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.