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arXiv · 2509.09098

Mountain Pass Critical Points of the Volume Constrained Area Functional

Gregory R. Chambers·Jared Marx-Kuo
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Abstract

We construct mountain pass critical points of the perimeter functional on sets of fixed volume. For a generic metric, this gives rise to a smooth almost embedded hypersurface with non-zero constant mean curvature. Our work utilizes recent techniques of Mazurwoski--Zhou \cite{mazurowski2024infinitely}, and a new result on the connectedness of Cacciopoli sets: any two smooth Cacciopoli sets can be connected by an $\mathbf{F}$-continuous map.

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Gregory R. Chambers, Jared Marx-Kuo. 2025-09-11. Mountain Pass Critical Points of the Volume Constrained Area Functional. https://arxiv.org/abs/2509.09098

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