Maps from Riemannian manifolds into non-degenerate Euclidean cones

Details Maps from Riemannian manifolds into non-degenerate Euclidean cones Maps from Riemannian manifolds into non-degenerate Euclidean cones

2010-03-30

arxiv.org/abs/1003.5772v2

Rev. Mat. Iberoamericana 26 (2010), no. 3, 1083-1100

Mathematics

Differential Geometry

19 pages, to appear

Scientific paper

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

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