Existence of embedded minimal tori in three-spheres with positive Ricci curvature

In this joint work with Xingzhe Li, we prove the strong Morse inequalities for the area functional in the space of embedded tori and spheres in the three sphere. As a consequence, we prove that in the three dimensional sphere with positive Ricci curvature, there exist at least 4 distinct embedded minimal tori. Suppose in addition that the metric is bumpy, then the three-sphere contains at least 9 distinct embedded minimal tori.