Yau's conjecture on infinitely many minimal surfaces

Remarks

Shing-Tung Yau asked in his 1982 list of problems in differential geometry whether every closed $3$-manifold contains infinitely many immersed minimal surfaces. The statement above, which gives embedded minimal hypersurfaces in all dimensions from $3$ to $7$, was proved by Antoine Song, building on work of Marques and Neves.

The proofs use Almgren–Pitts min-max theory for $k$-parameter sweepouts. Marques and Neves showed that the min-max values $\omega_k(M)$ for $k = 1, 2, \dots$ are areas of minimal hypersurfaces with multiplicities; by results of Gromov and Guth they grow like $k^{1/(n+1)}$. Under positive Ricci curvature this growth forces infinitely many distinct hypersurfaces, which gave the first cases of the conjecture. Liokumovich, Marques and Neves proved a Weyl law for the $\omega_k$. With it, Irie, Marques and Neves showed that for generic metrics the union of the closed embedded minimal hypersurfaces is dense in $M$. Song treated arbitrary metrics with a min-max theory on non-compact manifolds obtained from pieces of $M$ bounded by stable minimal hypersurfaces.

Example: in the round sphere $S^3$ the great $2$-spheres are infinitely many embedded minimal surfaces; the theorem asserts the same for every metric on every closed $3$-manifold.

Going deeper

For a student

The theorem is the analogue for minimal hypersurfaces of the existence of infinitely many closed geodesics on surfaces. Its proof is a showcase of Almgren–Pitts theory: the sequence of min-max widths behaves like the spectrum of the Laplacian, with a Weyl-type growth law. The dimension restriction comes from regularity: in dimension $8$ and higher, min-max hypersurfaces may have singular points.