Some results in the variational theory of the area and related functionals

Abstract

In this thesis, we prove some results in the variational theory of the minimal hypersurfaces, constant mean curvature (CMC) hypersurfaces and the Allen-Cahn equation. In Chapter 1, we summarize the main results. In Chapter 2, we show that the number of closed $c$-CMC hypersurfaces in a closed Riemannian manifold tends to infinity as $c$ tends to $0^+$. In Chapter 3, we show that the space of closed singular minimal hypersurfaces (in a closed Riemannian manifold), whose areas are uniformly bounded from above and the $p$-th Jacobi eigenvalues are uniformly bounded from below, is sequentially compact. In Chapter 4, we prove a sub-additive inequality for the volume spectrum of a closed Riemannian manifold. In Chapter 5, we prove two results related to the question to what extent the Almgren-Pitts min-max theory and the Allen-Cahn min-max theory agree. In Chapter 6, we prove the existence of finite energy min-max solutions to the Allen-Cahn equation on a complete Riemannian manifold of finite volume.