Existence of infinitely many free boundary minimal hypersurfaces

Abstract

In this paper, we prove that in any compact Riemannian manifold with smooth boundary, of dimension at least $3$ and at most $7$, there exist infinitely many almost properly embedded free boundary minimal hypersurfaces. This settles the free boundary version of Yau’s conjecture. The proof uses adaptions of A. Song’s work and the early works by Marques–Neves in their resolution to Yau’s conjecture, together with Li–Zhou’s regularity theorem for free boundary min-max minimal hypersurfaces.