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The trapping property of totally geodesic hyperplanes in Hadamard spaces. (English) Zbl 0865.53028

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 [V. Bangert and the author, Comment. Math. Helv. 71, No. 1, 122-143 (1996; Zbl 0856.53047)].

MSC:

53C20 Global Riemannian geometry, including pinching
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C65 Integral geometry

Citations:

Zbl 0856.53047
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References:

[1] V. Bangert, Laminations of 3-tori by least area surfaces, in ”Analysis et cetera” (P.H. Rabinowitz, E. Zehnder, eds.), Academic Press, Boston, 1990, 85–114.
[2] V. Bangert, U. Lang, Trapping quasiminimizing submanifolds in spaces of negative curvature, Comment. Math. Helv. 71 (1996), 122–143. · Zbl 0856.53047 · doi:10.1007/BF02566412
[3] H. Federer, Geometric Measure Theory, Springer Verlag, Berlin, 1969. · Zbl 0176.00801
[4] H. Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970), 767–771. · Zbl 0194.35803 · doi:10.1090/S0002-9904-1970-12542-3
[5] M. Gromov, Hyperbolic groups, in ”Essays in Group Theory” (S.M. Gersten, ed.), MSRI Publ. 8, Springer (1987), 75–263.
[6] M. Gromov, Foliated plateau problem, Part 1: Minimal varieties, GAFA 1 (1991), 14–79. · Zbl 0768.53011 · doi:10.1007/BF01895417
[7] G.A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. Math. 33 (1932), 719–739. · JFM 58.1256.01 · doi:10.2307/1968215
[8] W. Klingenberg, Geodätischer Fluss auf Mannigfaltigkeiten vom hyperbolischen Typ, Invent. Math. 14 (1971), 63–82. · Zbl 0219.53037 · doi:10.1007/BF01418743
[9] U. Lang, Quasi-minimizing surfaces in hyperbolic space, Math. Z. 210 (1992), 581–592. · Zbl 0758.53031 · doi:10.1007/BF02571815
[10] U. Lang, The existence of complete minimizing hypersurfaces in hyperbolic manifolds, Int. J. Math. 6 (1995), 45–58. · Zbl 0860.53002 · doi:10.1142/S0129167X95000055
[11] F. Morgan, Geometric Measure Theory. A Beginner’s Guide, Academic Press, Boston, 1988. · Zbl 0671.49043
[12] M. Morse, A fundamental class of geodesics on any surface of genus greater than one, Trans. Amer. Math. Soc. 26 (1924), 25–60. · JFM 50.0466.04 · doi:10.1090/S0002-9947-1924-1501263-9
[13] J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. Henri Poincaré, Analyse non linéaire 3 (1986), 229–272.
[14] J. Moser, A stability theorem for minimal foliations on a torus, Ergod. Th. & Dynam. Sys. 8 (1988), 251–281. · Zbl 0632.57018 · doi:10.1017/S0143385700009457
[15] L. Simon, Lectures on Geometric Measure Theory, Proc. Cent. Math. Anal. 3, Austr. Nat. U. Canberra, 1983. · Zbl 0546.49019
[16] L. Simon, A strict maximum principle for area minimizing hypersurfaces, J. Diff. Geom. 26 (1987), 327–335. · Zbl 0625.53052
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