Open Access
September, 2010 Maps from Riemannian manifolds into non-degenerate Euclidean cones
Luciano Mari , Marco Rigoli
Rev. Mat. Iberoamericana 26(3): 1057-1074 (September, 2010).


Let $M$ be a connected, non-compact $m$-dimensional Riemannian manifold. In this paper we consider smooth maps $\varphi: M \rightarrow \mathbb{R}^n$ 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 $\varphi$ and a metric parameter. As a side product, we recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case $\varphi$ is an isometric immersion, we also show that, if $M$ is sufficiently well-behaved and has non-positive sectional curvature, $\varphi(M)$ cannot be contained into a non-degenerate cone of $\mathbb{R}^{2m-1}$.


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Luciano Mari . Marco Rigoli . "Maps from Riemannian manifolds into non-degenerate Euclidean cones." Rev. Mat. Iberoamericana 26 (3) 1057 - 1074, September, 2010.


Published: September, 2010
First available in Project Euclid: 27 August 2010

zbMATH: 1205.53006
MathSciNet: MR2789376

Primary: 35B50 , 53C42
Secondary: 53C21

Keywords: Harmonic Maps , isometric immersion , maximum principles , Riemannian manifold

Rights: Copyright © 2010 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.26 • No. 3 • September, 2010
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