A mountain pass theorem for minimal hypersurfaces with fixed boundary

In this talk, we will be concerned with the existence of a third embedded minimal hypersurface spanning a closed submanifold B contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound B. For this purpose, we will study min-max methods similar to those of the work of De Lellis and Ramic, adapted to the discrete setting of Almgren and Pitts. It is expected that these methods can be applied in the construction of unbounded minimal surfaces in certain asymptotically flat 3-manifolds. We give some evidence that this can actually be done by analyzing related aspects in the case of the exact Riemannian Schwarzschild metric.