S.-T. Yau's four minimal two-spheres conjecture

Let $M$ be a manifold diffeomorphic to $S^3$, equipped with a Riemannian metric. An embedded minimal two-sphere in $M$ is an embedded minimal submanifold diffeomorphic to $S^2$. S.-T. Yau's conjecture. There exist four distinct embedded minimal two-spheres in $M$. Wang--Zhou confirmed this conjecture for Riemannian three-spheres when the metric is bumpy or has positive Ricci curvature; the paper proves a quantitative version under positive Ricci curvature and a positive scalar-curvature lower bound, so the unrestricted statement is no longer open in the cases covered by those results.