On the Moser's Bernstein Theorem
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913306562789376 |
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| author | Sampaio, José Edson da Silva, Euripedes Carvalho |
| author_facet | Sampaio, José Edson da Silva, Euripedes Carvalho |
| contents | In this paper, we prove the following version of the famous Bernstein's theorem: Let $X\subset \mathbb R^{n+k}$ be a closed and connected set with Hausdorff dimension $n$. Assume that $X$ satisfies the monotonicity formula at $p\in X$. Then, the following statements are equivalent:
(1) $X$ is an affine linear subspace;
(2) $X$ is a definable set that is Lipschitz regular at infinity and its geometric tangent cone at infinity, $C(X,\infty)$, is a linear subspace;
(3) $X$ is a definable set, blow-spherical regular at infinity and $C(X,\infty)$ is a linear subspace;
(4) $X$ is a definable set that is Lipschitz normally embedded at infinity and $C(X,\infty)$ is a linear subspace;
(5) the density of $X$ at infinity is 1.
Consequently, we prove the following generalization of Bernstein's theorem: Let $X\subset \mathbb R^{n+1}$ be a closed and connected set with Hausdorff dimension $n$. Assume that $X$ satisfies the monotonicity formula at $p\in X$ and there are compact sets $K\subset \mathbb{R}^n$ and $\tilde K\subset \mathbb{R}^{n+1}$ such that $X\setminus \tilde K$ is a minimal hypersurface that is the graph of a $C^2$-smooth function $u\colon \mathbb{R}^{n}\setminus K\to \mathbb{R}$. Assume that $u$ has bounded derivative whenever $n>7$. Then $X$ is a hyperplane. Several other results are also presented. For example, we generalize the o-minimal Chow's theorem, we prove that any entire complex analytic set that is bi-Lipschitz homeomorphic to a definable set in an o-minimal structure must be an algebraic set. We also obtain that Yau's Bernstein Problem, which says that an oriented stable complete minimal hypersurface in $\mathbb{R}^{n+1}$ with $n\leq 6$ must be a hyperplane, holds true whether the hypersurface is a definable set in an o-minimal structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01141 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Moser's Bernstein Theorem Sampaio, José Edson da Silva, Euripedes Carvalho Differential Geometry Algebraic Geometry Complex Variables 53A10, 53A07 (primary), 14J17, 53C42, 14P10 (secondary) In this paper, we prove the following version of the famous Bernstein's theorem: Let $X\subset \mathbb R^{n+k}$ be a closed and connected set with Hausdorff dimension $n$. Assume that $X$ satisfies the monotonicity formula at $p\in X$. Then, the following statements are equivalent: (1) $X$ is an affine linear subspace; (2) $X$ is a definable set that is Lipschitz regular at infinity and its geometric tangent cone at infinity, $C(X,\infty)$, is a linear subspace; (3) $X$ is a definable set, blow-spherical regular at infinity and $C(X,\infty)$ is a linear subspace; (4) $X$ is a definable set that is Lipschitz normally embedded at infinity and $C(X,\infty)$ is a linear subspace; (5) the density of $X$ at infinity is 1. Consequently, we prove the following generalization of Bernstein's theorem: Let $X\subset \mathbb R^{n+1}$ be a closed and connected set with Hausdorff dimension $n$. Assume that $X$ satisfies the monotonicity formula at $p\in X$ and there are compact sets $K\subset \mathbb{R}^n$ and $\tilde K\subset \mathbb{R}^{n+1}$ such that $X\setminus \tilde K$ is a minimal hypersurface that is the graph of a $C^2$-smooth function $u\colon \mathbb{R}^{n}\setminus K\to \mathbb{R}$. Assume that $u$ has bounded derivative whenever $n>7$. Then $X$ is a hyperplane. Several other results are also presented. For example, we generalize the o-minimal Chow's theorem, we prove that any entire complex analytic set that is bi-Lipschitz homeomorphic to a definable set in an o-minimal structure must be an algebraic set. We also obtain that Yau's Bernstein Problem, which says that an oriented stable complete minimal hypersurface in $\mathbb{R}^{n+1}$ with $n\leq 6$ must be a hyperplane, holds true whether the hypersurface is a definable set in an o-minimal structure. |
| title | On the Moser's Bernstein Theorem |
| topic | Differential Geometry Algebraic Geometry Complex Variables 53A10, 53A07 (primary), 14J17, 53C42, 14P10 (secondary) |
| url | https://arxiv.org/abs/2312.01141 |