In the early 1980s, S. T. Yau conjectured that any compact Riemannian
three-manifold admits an infinite number of closed immersed minimal
surfaces. We use min–max theory for the area functional to prove this conjecture
in the positive Ricci curvature setting. More precisely, we show that every
compact Riemannian manifold with positive Ricci curvature and dimension
at most seven contains infinitely many smooth, closed, embedded minimal
hypersurfaces. In the last section we mention some open problems related
with the geometry of these minimal hypersurfaces.