Maps from Riemannian manifolds into non-degenerate Euclidean cones. (English) Zbl 1205.53006

Mari, Luciano; Rigoli, Marco

Rev. Mat. Iberoam. 26, No. 3, 1057-1074 (2010).

Summary: Let \(M\) be a connected, non-compact \(m\)-dimensional Riemannian manifold. 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. 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 in a non-degenerate cone of \(\mathbb{R}^{2m-1}\).

MSC:

53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
35B50	Maximum principles in context of PDEs
53C43	Differential geometric aspects of harmonic maps

Keywords:

maximum principles; harmonic maps; isometric immersion; Riemannian manifold

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