Maps from Riemannian manifolds into non-degenerate Euclidean cones
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2010
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| _version_ | 1866916435549224960 |
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| author | Mari, Luciano Rigoli, Marco |
| author_facet | Mari, Luciano Rigoli, Marco |
| contents | Let $M$ be a connected, non-compact $m$-dimensional Riemannian manifold. In this paper we consider smooth maps $ϕ: M \to \mathbb{R}^n$ with images inside a non-degenerate cone. Under quite general assumptions on $M$, we provide a lower bound for the width of the cone in terms of the energy and the tension of $ϕ$ and a metric parameter. As a side product, we recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case $ϕ$ is an isometric immersion, we also show that, if $M$ is sufficiently well-behaved and has non-positive sectional curvature, $ϕ(M)$ cannot be contained into a non-degenerate cone of $\mathbb{R}^{2m-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1003_5772 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | Maps from Riemannian manifolds into non-degenerate Euclidean cones Mari, Luciano Rigoli, Marco Differential Geometry 53C42 (Primary), 35B50, 53C21 (Secondary) Let $M$ be a connected, non-compact $m$-dimensional Riemannian manifold. In this paper we consider smooth maps $ϕ: M \to \mathbb{R}^n$ with images inside a non-degenerate cone. Under quite general assumptions on $M$, we provide a lower bound for the width of the cone in terms of the energy and the tension of $ϕ$ and a metric parameter. As a side product, we recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case $ϕ$ is an isometric immersion, we also show that, if $M$ is sufficiently well-behaved and has non-positive sectional curvature, $ϕ(M)$ cannot be contained into a non-degenerate cone of $\mathbb{R}^{2m-1}$. |
| title | Maps from Riemannian manifolds into non-degenerate Euclidean cones |
| topic | Differential Geometry 53C42 (Primary), 35B50, 53C21 (Secondary) |
| url | https://arxiv.org/abs/1003.5772 |