Nonlocal minimal surfaces are unique for all but countably many parameters in any strictly increasing family of exterior data, and arbitrarily small perturbations yield smooth minimizers in one dimension beyond the critical threshold.
2308.13209 , archivePrefix=
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abstract
In this work, we show the generic uniqueness of minimizers for a large class of energies, including the Alt-Caffarelli and Alt-Phillips functionals. We then prove the generic regularity of free boundaries for minimizers of the one-phase Alt-Caffarelli and Alt-Phillips functionals, for a monotone family of boundary data $\{\varphi_t\}_{t\in(-1,1)}$. More precisely, we show that for a co-countable subset of $\{\varphi_t\}_{t\in(-1,1)}$, minimizers have smooth free boundaries in $\mathbb{R}^5$ for the Alt-Caffarelli and in $\mathbb{R}^3$ for the Alt-Phillips functional. In general dimensions, we show that the singular set is one dimension smaller than expected for almost every boundary datum in $\{\varphi_t\}_{t\in(-1,1)}$.
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2026 3representative citing papers
Intrinsic and extrinsic area density bounds are equivalent for complete connected smooth minimal immersions in Euclidean space of any dimension and codimension, enabling extension of Schoen-Simon-Yau estimates to n=6.
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Nonlocal minimal surfaces are generically unique, and smooth in one extra dimension
Nonlocal minimal surfaces are unique for all but countably many parameters in any strictly increasing family of exterior data, and arbitrarily small perturbations yield smooth minimizers in one dimension beyond the critical threshold.
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Equivalence of intrinsic and extrinsic area bounds for minimal surfaces
Intrinsic and extrinsic area density bounds are equivalent for complete connected smooth minimal immersions in Euclidean space of any dimension and codimension, enabling extension of Schoen-Simon-Yau estimates to n=6.
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Improvement of flatness in annuli
A PDE-based improvement-of-flatness technique for annuli provides an alternative proof of the end-structure and asymptotics for finite Morse index minimal hypersurfaces with Euclidean area growth in low dimensions.