Let be a connected, non-compact -dimensional Riemannian manifold. In this paper we consider smooth maps with images inside a non-degenerate cone. Under quite general assumptions on , we provide a lower bound for the width of the cone in terms of the energy and the tension of and a metric parameter. As a side product, we recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case is an isometric immersion, we also show that, if is sufficiently well-behaved and has non-positive sectional curvature, cannot be contained into a non-degenerate cone of .
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Luciano Mari, Marco Rigoli, Maps from Riemannian manifolds into non-degenerate Euclidean cones. Rev. Mat. Iberoam. 26 (2010), no. 3, pp. 1057–1074DOI 10.4171/RMI/627