Compactness and generic finiteness for free boundary minimal hypersurfaces (I)

Abstract

Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical convergence away from finitely many points. We show that the limit of a sequence of such hypersurfaces always inherits a non-trivial Jacobi field when it has multiplicity one. In a forthcoming paper, we will construct Jacobi fields when the convergence has higher multiplicity.

arXiv commentWe fixed a gap in the construction of Jacobi fields and split the original paper into two parts. The followup paper(the second part) will deal with the construction of Jacobi fields when the convergence has higher multiplicity

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Cited by in corpus (8)

• Free boundary minimal surfaces in the unit ball : recent advances and open questions
• Min-max theory for free boundary minimal hypersurfaces II -- General Morse index bounds and applications
• Existence of infinitely many free boundary minimal hypersurfaces
• The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space
• Multiplicity one for min-max theory in compact manifolds with boundary and its applications
• A two-piece property for free boundary minimal hypersurfaces in the $(n+1)$-dimensional ball
• Generic density of equivariant min-max hypersurfaces
• Bubbling analysis and geometric convergence results for free boundary minimal surfaces