Spectral Bernstein theorems for submanifolds in Euclidean spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914583400153088 |
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| author | Dong, Yuxin Lin, Hezi Zhang, Wei |
| author_facet | Dong, Yuxin Lin, Hezi Zhang, Wei |
| contents | In this paper, we consider the essential spectrum of submanifolds in Euclidean spaces under various geometric hypotheses. Our results involve extrinsic conditions such as finite total mean curvature, the convergence of the gradient of the extrinsic distance, and the extrinsic volume growth or the pinching curvature. In particular, we prove that the essential spectrum of a complete non-compact submanifold $M^n$ in a Euclidean space is $[0, +\infty)$ provided the second fundamental form $A$ of $M^n$ satisfies $\|A\|_{L^p} < \infty$, $p>n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21370 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral Bernstein theorems for submanifolds in Euclidean spaces Dong, Yuxin Lin, Hezi Zhang, Wei Differential Geometry In this paper, we consider the essential spectrum of submanifolds in Euclidean spaces under various geometric hypotheses. Our results involve extrinsic conditions such as finite total mean curvature, the convergence of the gradient of the extrinsic distance, and the extrinsic volume growth or the pinching curvature. In particular, we prove that the essential spectrum of a complete non-compact submanifold $M^n$ in a Euclidean space is $[0, +\infty)$ provided the second fundamental form $A$ of $M^n$ satisfies $\|A\|_{L^p} < \infty$, $p>n$. |
| title | Spectral Bernstein theorems for submanifolds in Euclidean spaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.21370 |