Maps from Riemannian manifolds into non-degenerate Euclidean cones

TL;DR

This paper establishes bounds on the geometry of maps from Riemannian manifolds into Euclidean cones, linking energy, tension, and curvature, and recovers known results about harmonic maps and minimal immersions.

Contribution

It provides new lower bounds for cone widths based on map properties and extends classical results to broader geometric contexts.

Findings

Lower bounds for cone width in terms of energy and tension.

Recovery of classical results on harmonic maps and minimal immersions.

Non-existence of certain immersions into cones under curvature conditions.

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