On the spectrum of bounded immersions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2012
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| _version_ | 1866929538768830464 |
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| author | Bessa, Gregorio Pacelli Jorge, Luquesio P. Mari, Luciano |
| author_facet | Bessa, Gregorio Pacelli Jorge, Luquesio P. Mari, Luciano |
| contents | In this paper, we investigate the relationship between the discreteness of the spectrum of a non-compact, extrinsically bounded submanifold $φ\colon M^m \ra N^n$ and the Hausdorff dimension of its limit set $\limφ$. In particular, we prove that if $φ\colon \!M^2 \ra D \subseteq \R^3$ is a minimal immersion into an open, bounded, strictly convex subset $D$ with $C^2$-boundary, then $M$ has discrete spectrum provided that $\haus_Ψ(\limφ\cap D)=0$, where $\haus_Ψ$ is the generalized Hausdorff measure of order $Ψ(t) = t^2|\log t|$. Our theorem applies to a number of examples recently constructed by various authors in the light of N. Nadirashvili's discovery of complete, bounded minimal disks in $\R^3$, as well as to solutions of Plateau's problems, giving a fairly complete answer to a question posed by S.T. Yau in his Millenium Lectures. Suitable counter-examples show the sharpness of our results: in particular, we develop a simple criterion for the existence of essential spectrum which is suited for the techniques developed after Jorge-Xavier and Nadirashvili's examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1211_6059 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | On the spectrum of bounded immersions Bessa, Gregorio Pacelli Jorge, Luquesio P. Mari, Luciano Differential Geometry 53C42, 53A10, 58C40, 58E12 In this paper, we investigate the relationship between the discreteness of the spectrum of a non-compact, extrinsically bounded submanifold $φ\colon M^m \ra N^n$ and the Hausdorff dimension of its limit set $\limφ$. In particular, we prove that if $φ\colon \!M^2 \ra D \subseteq \R^3$ is a minimal immersion into an open, bounded, strictly convex subset $D$ with $C^2$-boundary, then $M$ has discrete spectrum provided that $\haus_Ψ(\limφ\cap D)=0$, where $\haus_Ψ$ is the generalized Hausdorff measure of order $Ψ(t) = t^2|\log t|$. Our theorem applies to a number of examples recently constructed by various authors in the light of N. Nadirashvili's discovery of complete, bounded minimal disks in $\R^3$, as well as to solutions of Plateau's problems, giving a fairly complete answer to a question posed by S.T. Yau in his Millenium Lectures. Suitable counter-examples show the sharpness of our results: in particular, we develop a simple criterion for the existence of essential spectrum which is suited for the techniques developed after Jorge-Xavier and Nadirashvili's examples. |
| title | On the spectrum of bounded immersions |
| topic | Differential Geometry 53C42, 53A10, 58C40, 58E12 |
| url | https://arxiv.org/abs/1211.6059 |