Exploring new extrinsic upper bounds on the first eigenvalue of the Laplace operator for compact submanifolds in Euclidean spaces
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911852369281024 |
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| author | Palomo, Francisco J. Romero, Alfonso |
| author_facet | Palomo, Francisco J. Romero, Alfonso |
| contents | Upper bounds of the first non-trivial eigenvalue $λ_1$ of the Laplace operator of a compact submanifold $M^n$ of Euclidean space $\R^{m+1}$, by means of a new technique, are obtained. Each of the upper bounds of $λ_1$ depends on the length of mean curvature vector field, the dimension $n$, the volume of $M^n$, and of a vector of $\R^{m+1}$. When $M^n$ does not lie minimally in a hypersphere of $\R^{m+1}$, classical Reilly's inequality \cite{Re} is improved and new upper bounds are explicitly computed. For instance, considering a torus of revolution whose generating circle has a radius of $1$ and is centered at distance $\sqrt{2}$ from the axis of revolution, we find $λ_1 < \frac{4}{3}(\sqrt{2}-1)\approx 0.552284$, whereas Reilly's upper bound gives $λ_1 < 1/\sqrt{2}\approx 0.707106$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05277 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exploring new extrinsic upper bounds on the first eigenvalue of the Laplace operator for compact submanifolds in Euclidean spaces Palomo, Francisco J. Romero, Alfonso Differential Geometry Primary 53C40, 35P15. Secondary 53C42, 58J50 Upper bounds of the first non-trivial eigenvalue $λ_1$ of the Laplace operator of a compact submanifold $M^n$ of Euclidean space $\R^{m+1}$, by means of a new technique, are obtained. Each of the upper bounds of $λ_1$ depends on the length of mean curvature vector field, the dimension $n$, the volume of $M^n$, and of a vector of $\R^{m+1}$. When $M^n$ does not lie minimally in a hypersphere of $\R^{m+1}$, classical Reilly's inequality \cite{Re} is improved and new upper bounds are explicitly computed. For instance, considering a torus of revolution whose generating circle has a radius of $1$ and is centered at distance $\sqrt{2}$ from the axis of revolution, we find $λ_1 < \frac{4}{3}(\sqrt{2}-1)\approx 0.552284$, whereas Reilly's upper bound gives $λ_1 < 1/\sqrt{2}\approx 0.707106$. |
| title | Exploring new extrinsic upper bounds on the first eigenvalue of the Laplace operator for compact submanifolds in Euclidean spaces |
| topic | Differential Geometry Primary 53C40, 35P15. Secondary 53C42, 58J50 |
| url | https://arxiv.org/abs/2401.05277 |