The trapping property of totally geodesic hyperplanes in Hadamard spaces
The main result of this paper is: For all \(Q\), \(a\geq 1\) and integers \(n>k\geq 2\) there exists a (computable) constant \(D\leq 0\) such that the following holds. Let \(X=X'\times\mathbb{R}^{n-k}\), where \(X\) is either hyperbolic \(k\)-space \(\mathbb{H}^k\) or a Hadamard manifold of dimension \(k=2\) with curvature \(-a^2\leq K\leq -1\). Let \(G=G'\times\mathbb{R}^{n-k}\subset X\) for some totally geodesic \((k-1)\)-plane \(G'\subset X'\). If \(S\in R^{\text{loc}}_{n-1}(X)\) (=space of locally rectifiable \((n-1)\)-currents) is \(Q\)-minimizing in \(X\), \(\text{spt }S\) lies within finite distance from \(G\), and \(\text{spt }\partial S\subset G\), then \(\text{spt }S\) lies within distance \(D\) from \(G\). In addition to this, an analogue trapping property is proved for sufficiently small quasiminimality constant \(Q\) in the case \(X=X'\times Y\), \(Y\) with cocompact isometry group. The results rely on [\textit{V. Bangert} and the author, Comment. Math. Helv. 71, No. 1, 122-143 (1996; Zbl 0856.53047)].
totally geodesic hyperplanes in Hadamard spaces
scientific article; zbMATH DE number 934960