Revista Matemática Iberoamericana

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Volume 26, Issue 3, 2010, pp. 1057–1074
DOI: 10.4171/RMI/627

Published online: 2010-12-31

Maps from Riemannian manifolds into non-degenerate Euclidean cones

Luciano Mari[1] and Marco Rigoli[2]

(1) Università di Milano, Italy
(2) Università di Milano, Italy

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

Keywords: Maximum principles, harmonic maps, isometric immersion, Riemannian manifold

Mari Luciano, Rigoli Marco: Maps from Riemannian manifolds into non-degenerate Euclidean cones. Rev. Mat. Iberoam. 26 (2010), 1057-1074. doi: 10.4171/RMI/627