Maps from Riemannian manifolds into non-degenerate Euclidean cones

Abstract

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