Maps from Riemannian manifolds into non-degenerate Euclidean cones

  • Luciano Mari

    Università di Milano, Italy
  • Marco Rigoli

    Università di Milano, Italy


Let MM be a connected, non-compact mm-dimensional Riemannian manifold. In this paper we consider smooth maps φ:MRn\varphi: M \rightarrow \mathbb{R}^n with images inside a non-degenerate cone. Under quite general assumptions on MM, we provide a lower bound for the width of the cone in terms of the energy and the tension of φ\varphi and a metric parameter. As a side product, we recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case φ\varphi is an isometric immersion, we also show that, if MM is sufficiently well-behaved and has non-positive sectional curvature, φ(M)\varphi(M) cannot be contained into a non-degenerate cone of R2m1\mathbb{R}^{2m-1}.

Cite this article

Luciano Mari, Marco Rigoli, Maps from Riemannian manifolds into non-degenerate Euclidean cones. Rev. Mat. Iberoam. 26 (2010), no. 3, pp. 1057–1074

DOI 10.4171/RMI/627