Stable anisotropic minimal hypersurfaces in $\mathbf {R}^{4}$
We show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in $\mathbf {R}^4$ has intrinsic cubic volume growth, provided the parametric elliptic integral is $C^2$ -close to the area functional. We also obtain an interior volume upper bound for stable anisotrop...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2023-01-01
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| Series: | Forum of Mathematics, Pi |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S205050862300001X/type/journal_article |
| _version_ | 1811156329955328000 |
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| author | Otis Chodosh Chao Li |
| author_facet | Otis Chodosh Chao Li |
| author_sort | Otis Chodosh |
| collection | DOAJ |
| description | We show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in
$\mathbf {R}^4$
has intrinsic cubic volume growth, provided the parametric elliptic integral is
$C^2$
-close to the area functional. We also obtain an interior volume upper bound for stable anisotropic minimal hypersurfaces in the unit ball. We can estimate the constants explicitly in all of our results. In particular, this paper gives an alternative proof of our recent stable Bernstein theorem for minimal hypersurfaces in
$\mathbf {R}^4$
. The new proof is more closely related to techniques from the study of strictly positive scalar curvature. |
| first_indexed | 2024-04-10T04:48:35Z |
| format | Article |
| id | doaj.art-b74c876d146f4476aec343bd794c943e |
| institution | Directory Open Access Journal |
| issn | 2050-5086 |
| language | English |
| last_indexed | 2024-04-10T04:48:35Z |
| publishDate | 2023-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Pi |
| spelling | doaj.art-b74c876d146f4476aec343bd794c943e2023-03-09T12:34:19ZengCambridge University PressForum of Mathematics, Pi2050-50862023-01-011110.1017/fmp.2023.1Stable anisotropic minimal hypersurfaces in $\mathbf {R}^{4}$Otis Chodosh0https://orcid.org/0000-0002-6124-7889Chao Li1https://orcid.org/0000-0003-2735-9139Department of Mathematics, Stanford University, Building 380, Stanford, CA 94305, USA; E-mail:Courant Institute, New York University, 251 Mercer St, New York, NY 10012, USAWe show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in $\mathbf {R}^4$ has intrinsic cubic volume growth, provided the parametric elliptic integral is $C^2$ -close to the area functional. We also obtain an interior volume upper bound for stable anisotropic minimal hypersurfaces in the unit ball. We can estimate the constants explicitly in all of our results. In particular, this paper gives an alternative proof of our recent stable Bernstein theorem for minimal hypersurfaces in $\mathbf {R}^4$ . The new proof is more closely related to techniques from the study of strictly positive scalar curvature.https://www.cambridge.org/core/product/identifier/S205050862300001X/type/journal_article53C4235J5053A1049F10 |
| spellingShingle | Otis Chodosh Chao Li Stable anisotropic minimal hypersurfaces in $\mathbf {R}^{4}$ Forum of Mathematics, Pi 53C42 35J50 53A10 49F10 |
| title | Stable anisotropic minimal hypersurfaces in $\mathbf {R}^{4}$ |
| title_full | Stable anisotropic minimal hypersurfaces in $\mathbf {R}^{4}$ |
| title_fullStr | Stable anisotropic minimal hypersurfaces in $\mathbf {R}^{4}$ |
| title_full_unstemmed | Stable anisotropic minimal hypersurfaces in $\mathbf {R}^{4}$ |
| title_short | Stable anisotropic minimal hypersurfaces in $\mathbf {R}^{4}$ |
| title_sort | stable anisotropic minimal hypersurfaces in mathbf r 4 |
| topic | 53C42 35J50 53A10 49F10 |
| url | https://www.cambridge.org/core/product/identifier/S205050862300001X/type/journal_article |
| work_keys_str_mv | AT otischodosh stableanisotropicminimalhypersurfacesinmathbfr4 AT chaoli stableanisotropicminimalhypersurfacesinmathbfr4 |