Density and spectrum of minimal submanifolds in space forms
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arXiv
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| Format: | Preprint |
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2014
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| author | Lima, Barnabé Pessoa Montenegro, José Fabio Mari, Luciano Vieira, Franciane B. |
| author_facet | Lima, Barnabé Pessoa Montenegro, José Fabio Mari, Luciano Vieira, Franciane B. |
| contents | Let $M^m$ be a minimal properly immersed submanifold in an ambient space close, in a suitable sense, to the space form $\mathbb{N}^n_k$ of curvature $-k\le 0$. In this paper, we are interested in the relation between the density function $Θ(r)$ of $M^m$ and the spectrum of the Laplace-Beltrami operator. In particular, we prove that if $Θ(r)$ has subexponential growth (when $k<0$) or sub-polynomial growth ($k=0$) along a sequence, then the spectrum of $M^m$ is the same as that of the space form $\mathbb{N}^m_k$. Notably, the result applies to Anderson's (smooth) solutions of Plateau's problem at infinity on the hyperbolic space $\mathbb{H}^n$, independently of their boundary regularity. We also give a simple condition on the second fundamental form that ensures $M$ to have finite density. In particular, we show that minimal submanifolds of $\mathbb{H}^n$ with finite total curvature have finite density. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1407_5280 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Density and spectrum of minimal submanifolds in space forms Lima, Barnabé Pessoa Montenegro, José Fabio Mari, Luciano Vieira, Franciane B. Differential Geometry 58J50 (Primary), 58C21 (Secondary), 35P15 Let $M^m$ be a minimal properly immersed submanifold in an ambient space close, in a suitable sense, to the space form $\mathbb{N}^n_k$ of curvature $-k\le 0$. In this paper, we are interested in the relation between the density function $Θ(r)$ of $M^m$ and the spectrum of the Laplace-Beltrami operator. In particular, we prove that if $Θ(r)$ has subexponential growth (when $k<0$) or sub-polynomial growth ($k=0$) along a sequence, then the spectrum of $M^m$ is the same as that of the space form $\mathbb{N}^m_k$. Notably, the result applies to Anderson's (smooth) solutions of Plateau's problem at infinity on the hyperbolic space $\mathbb{H}^n$, independently of their boundary regularity. We also give a simple condition on the second fundamental form that ensures $M$ to have finite density. In particular, we show that minimal submanifolds of $\mathbb{H}^n$ with finite total curvature have finite density. |
| title | Density and spectrum of minimal submanifolds in space forms |
| topic | Differential Geometry 58J50 (Primary), 58C21 (Secondary), 35P15 |
| url | https://arxiv.org/abs/1407.5280 |