Almgren–Pitts min-max theory

Summary

Almgren–Pitts min-max theory constructs closed minimal hypersurfaces in a closed Riemannian manifold by a min-max over families of hypersurfaces that sweep out the manifold. It is the higher-dimensional analogue of the Birkhoff min-max method for closed geodesics. Frederick Almgren set up the topology of spaces of cycles, and Jon Pitts proved the regularity of the min-max hypersurface in low dimensions. Fernando Marques and André Neves developed the theory further and used it to prove the Willmore conjecture. It is also the main tool behind Yau's conjecture on infinitely many minimal surfaces.