Min-max theory for minimal hypersurfaces with boundary

Abstract

In this thesis we present a result concerning existence and regularity of minimal surfaces with boundary in Riemannian manifolds obtained via a min-max construction, both in fixed and free boundary context. We start by considering a smooth, compact, oriented Riemmanian manifold ($\mathcal{M},\mathit{g}$) of dimension $n$ + 1 with a uniform convexity property (the principal curvatures of $\partial\mathcal{M}$ with respect to the inner normal have a uniform positive lower bound). In the spirit of an analogous approach presented in Colding-De Lellis (cf. [10]), we then construct generalized families of hypersurfaces in $\mathcal{M}$, with appropriate boundary conditions. To be more precise, in the fixed boundary setting this will basically mean that these hypersurfaces will have a fixed common boundary $\gamma$, which is an ($n$ - 1)-dimensional smooth, closed, oriented submanifold of $\partial\mathcal{M}$, and in the free boundary setting that their boundaries lie in $\partial\mathcal{M}$. Moreover, we will consider more general parameter spaces for these families. After constructing a suitable homotopy class of such families and assuming that it satisfies a certain gap, we can prove the existence of a nontrivial, embedded, minimal hypersurface (with a codimension 7 singular set) and corresponding boundary conditions. As a corollary, we consider a special case of two smooth, strictly stable surfaces bounding an open domain $A$ and meeting only in the common boundary $\gamma \subset \partial\mathcal{M}$, and show the existence of a homotopy class with the required energy gap. The main theorem then furnishes the existence of a third minimal surface.