Bellettini, Costante;
(2025)
Extensions of Schoen–Simon–Yau and Schoen–Simon theorems via iteration à la De Giorgi.
Inventiones mathematicae
, 240
(1)
pp. 1-34.
10.1007/s00222-025-01317-0.
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Abstract
We give an alternative proof of the Schoen–Simon–Yau curvature estimates and associated Bernstein-type theorems (Schoen et al. in Acta Math. 134:275–288, 1975), and extend the original result by including the case of 6-dimensional (stable minimal) immersions. The key step is an ε-regularity theorem, that assumes smallness of the scale-invariant L2 norm of the second fundamental form. Further, we obtain a graph description, in the Lipschitz multi-valued sense, for any stable minimal immersion of dimension n ≥ 2, that may have a singular set Σ of locally finite Hn−2-measure, and that is weakly close to a hyperplane. (In fact, if the Hn−2-measure of the singular set vanishes, the conclusion is strengthened to a union of smooth graphs.) This follows directly from an ε-regularity theorem, that assumes smallness of the scale-invariant L2 tilt-excess (verified when the hypersurface is weakly close to a hyperplane). Specialising the multi-valued decomposition to the case of embeddings, we recover the Schoen–Simon theorem (Schoen and Simon 34:741–797, 1981). In both ε-regularity theorems the relevant quantity (respectively, length of the second fundamental form and tilt function) solves a non-linear PDE on the immersed minimal hypersurface. The proof is carried out intrinsically (without linearising the PDE) by implementing an iteration method à la De Giorgi (from the linear De Giorgi–Nash–Moser theory). Stability implies estimates (intrinsic weak Caccioppoli inequalities) that make the iteration effective despite the non-linear framework. (In both ε-regularity theorems the method gives explicit constants that quantify the required smallness.)
| Type: | Article |
|---|---|
| Title: | Extensions of Schoen–Simon–Yau and Schoen–Simon theorems via iteration à la De Giorgi |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00222-025-01317-0 |
| Publisher version: | https://doi.org/10.1007/s00222-025-01317-0 |
| Language: | English |
| Additional information: | This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10203938 |
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