On minimal graphs of sublinear growth over manifolds with non-negative Ricci curvature
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917057177583616 |
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| author | Colombo, Giulio Mari, Luciano Rigoli, Marco |
| author_facet | Colombo, Giulio Mari, Luciano Rigoli, Marco |
| contents | We prove that entire solutions of the minimal hypersurface equation \[
\mathrm{div}\left(\frac{Du}{\sqrt{1+|Du|^2}}\right) = 0 \] on a complete manifold with $\mathrm{Ric} \ge 0$, whose negative part grows like $\mathcal{O}(r/\log r)$ ($r$ the distance from a fixed origin), are constant. This extends the Bernstein Theorem for entire positive minimal graphs established in recent years. The proof depends on a new technique to get gradient bounds by means of integral estimates, which does not require any further geometric assumption on $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_15620 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On minimal graphs of sublinear growth over manifolds with non-negative Ricci curvature Colombo, Giulio Mari, Luciano Rigoli, Marco Differential Geometry Analysis of PDEs 53C21, 53C42 (Primary) 53C24, 58J65, 31C12 (Secondary) We prove that entire solutions of the minimal hypersurface equation \[ \mathrm{div}\left(\frac{Du}{\sqrt{1+|Du|^2}}\right) = 0 \] on a complete manifold with $\mathrm{Ric} \ge 0$, whose negative part grows like $\mathcal{O}(r/\log r)$ ($r$ the distance from a fixed origin), are constant. This extends the Bernstein Theorem for entire positive minimal graphs established in recent years. The proof depends on a new technique to get gradient bounds by means of integral estimates, which does not require any further geometric assumption on $M$. |
| title | On minimal graphs of sublinear growth over manifolds with non-negative Ricci curvature |
| topic | Differential Geometry Analysis of PDEs 53C21, 53C42 (Primary) 53C24, 58J65, 31C12 (Secondary) |
| url | https://arxiv.org/abs/2310.15620 |