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 $\mathbb{R}^{2m-1}$.

Maps from Riemannian manifolds into non-degenerate Euclidean cones / L. Mari, M. Rigoli. - In: REVISTA MATEMATICA IBEROAMERICANA. - ISSN 0213-2230. - 26:3(2010 Sep), pp. 1057-1074.

Maps from Riemannian manifolds into non-degenerate Euclidean cones

L. Mari
Primo
;
M. Rigoli
Ultimo
2010-09

Abstract

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 $\mathbb{R}^{2m-1}$.
Harmonic maps; Isometric immersion; Maximum principles; Riemannian manifold
Settore MAT/03 - Geometria
Settore MAT/05 - Analisi Matematica
http://projecteuclid.org/euclid.rmi/1282913832
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/150470
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