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.

在这项工作中，我们证明存在先验已知的第三个嵌入的最小超曲面，该曲面覆盖了山口类型的闭合子流形\（\ gamma \），包含在具有凸边界的紧凑黎曼流形的边界中。两个严格稳定的最小超曲面以\（\ gamma \）为界。为此，我们开发了类似于De Lellis和Ramic（Ann。Inst。Fourier 68（5）：1909 –1986，2018）的最小-最大方法，适用于Almgren和Pitts的离散设置。我们的方法允许考虑其中边界为\（\ gamma \）的两个稳定超曲面在内部点相交的情况。