On the Gehring link problem and the isoperimetric inequality of Bombieri and Simon (Q1091620)

Here, by substituting the euclidean space \(R^{p+q+1}\) by a complete Riemannian manifold \(E^ n\), the author generalizes a result of Bombieri, Gage, Simon and others on an evaluation problem -- the Gehring link problem -- of the volume of compact oriented linked submanifolds \(M^ p\) and \(N^ q\) in the euclidean space \(R^{p+q+1}\), which are at a distance at least one from each other (superscripts denote the dimension). In order to meet the generalization, the notion of chain linked manifolds is introduced: A compact oriented submanifold \(M^ p\) of \(E^ n\) is said to be chain linked to a closed subset N (disjoint to \(M^ p\) and not necessarily to be a submanifold) of \(E^ n\), if \(M^ p\) is zero in \(H_ p({\mathbb{E}}^ n,{\mathbb{Z}})\) but not zero in \(H_ p(E^ n-N,Z)\). The chain linked condition replaces the evaluation problem by the proof of an isoperimetric inequality for the bounding current Y of \(M^ p\), the support of which should intersect with N, if \(M^ p\) and N are chain linked. Thus a computation yields the following theorem: Let \(E^ n\) be a complete Riemannian manifold whose sectional curvatures are \(<K<0\) and let inj denote the inclusion of a compact submanifold M into E. If the induced map \(inj_*\) on the fundamental group is trivial and if \(M^ p\) is chain linked to a closed set N of \(E^ n\), then it holds that \[ vol(M^ p)\geq vol(S^ p)S_ k(r)^ p \] where \(S_ K(r)\) is defined by \[ S_ K(r) = \begin{cases} r & K=0 \\ \sinh(\sqrt{| K|}r)/\sqrt{| K|} & K<0 \end{cases} \] for the distance r from M to N, and \(S^ p\) stands for p-sphere.