Inserted: 2 nov 2016
Last Updated: 17 jul 2018
Journal: To appear in Annales de l'Institut Fourier
In this note we propose a min-max theory for embedded hypersurfaces with a fixed boundary and apply it to prove several theorems about the existence of embedded minimal hypersurfaces with a given boundary. A simpler variant of these theorems holds also for the case of the free boundary minimal surfaces.