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May 2023 Stable minimal hypersurfaces in ℝ^N+1+ℓ with singular set an arbitrary closed K⊂{0}×ℝ^ℓ
Leon Simon
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Ann. of Math. (2) 197(3): 1205-1234 (May 2023). DOI: 10.4007/annals.2023.197.3.4


With respect to a $C^\infty$ metric which is close to the standard Euclidean metric on $\mathbb{R}^{N+1+\ell}$, where $N\ge 7$ and $\ell\ge 1$ are given, we construct a class of embedded $(N+\ell)$-dimensional hypersurfaces (without boundary) which are minimal and strictly stable, and which have singular set equal to an arbitrary preassigned closed subset $K\subset \{0\}\times \mathbb{R}^{\ell}$. Thus the question is settled, with a strong affirmative, as to whether there can be " gaps" or even fractional dimensional parts in the singular set. Such questions, for both stable and unstable minimal submanifolds, remain open in all dimensions in the case of real analytic metrics and, in particular, for the standard Euclidean metric.

The construction used here involves the analysis of solutions $u$ of the symmetric minimal surface equation on domains $\Omega\subset\mathbb{R}^{n}$ whose symmetric graphs (i.e., $\{(x,\xi)\in\Omega\times \mathbb{R}^{m}: |\xi| = u(x)\}$) lie on one side of a cylindrical minimal cone including, in particular, a Liouville type theorem for complete solutions (i.e., the case $\Omega=\mathbb{R}^{n}$).


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Leon Simon. "Stable minimal hypersurfaces in ℝ^N+1+ℓ with singular set an arbitrary closed K⊂{0}×ℝ^ℓ." Ann. of Math. (2) 197 (3) 1205 - 1234, May 2023.


Published: May 2023
First available in Project Euclid: 23 March 2023

Digital Object Identifier: 10.4007/annals.2023.197.3.4

Primary: 28A75 , 35A30 , 53C23

Keywords: minimal submanifold , singular set , strict stability

Rights: Copyright © 2023 Department of Mathematics, Princeton University


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Vol.197 • No. 3 • May 2023
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