Unboundedness of Horn with finite positive part of curvature in the Euclidean space (Q1058741)

A tube M is a complete Riemannian manifold homeomorphic to a 2-disc with the outer boundary included and the center excluded. Let \(g(M)=\inf \ell (\gamma)\), where \(\ell (\gamma)\) is the length of any loop homotopic to \(\partial M\). A tube is called a horn if any sequence of loops \(\gamma_ i\) homotopic to \(\partial M\) and \(\ell (\gamma_ i)\to g(M)\), diverges in the metric of M. Let \(f: M\to {\mathbb{R}}^ 3\) be an isometric immersion of a horn M such that \(\int_{M}K^+ dA<\infty\), where \(K^+\) is the positive part of the curvature. The author proves that under this condition f(M) is unbounded in \({\mathbb{R}}^ 3\).