A mountain pass theorem for minimal hypersurfaces with fixed boundary

Abstract

In this work, we prove the existence of a third embedded minimal hypersurface spanning a closed submanifold $$\gamma $$ , of mountain pass type, 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 $$\gamma $$ . In order to do so, we develop min–max methods similar to those of De Lellis and Ramic (Ann. Inst. Fourier 68(5): 1909 –1986, 2018) adapted to the discrete setting of Almgren and Pitts. Our approach allows one to consider the case in which the two stable hypersurfaces with boundary $$\gamma $$ intersect at interior points.