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Compactness analysis for free boundary minimal hypersurfaces. (English) Zbl 1414.53008

A hypersurface \(M^{n}\), \(n\geq 2\), is free-boundary minimal hypersurface in a compact Riemannian manifold with boundary \((N^{n+1}, g)\) if it is a critical point for the \(n\)-dimensional area functional under the unique constraint that \(\partial M \subset \partial N\) or equivalently if \(M\) has zero mean curvature and meets the ambient boundary orthogonally. The authors study compactness phenomena involving free-boundary minimal hypersurfaces in compact Riemannian manifold with boundary of dimension less than 8. They give natural geometric conditions that imply strong one-sheeted graphical subsequential convergence, and describe various relevant phenomena when instead multi-sheeted convergence occurs. They show that a uniform lower bound on some eigenvalue of the Jacobi operator together with a uniform upper bound on the area is sufficient for a weak compactness result, in the sense of graphical but possibly multi-sheeted convergence away from finitely many points where necks (or half-necks, at boundary points) may form. They investigate for \(n\geq 2\) when a properly embedded minimal hypersurface is the limit of free-boundary minimal hypersurfaces of a compact Riemannian manifold with boundary \((N^{n+1}, g)\).
The authors present two appendices that are devoted to certain technical aspects appearing in the proofs of some of their results. Namely in the first appendix they give a detailed derivation of the second variation formula for smooth hypersurfaces with boundary (without assuming either minimality of the hypersurface or orthogonal intersection with the ambient manifold).

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
49Q05 Minimal surfaces and optimization
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