Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-25T04:24:11.894Z Has data issue: false hasContentIssue false
  • English
  • Français

Polygons with Prescribed Gauss Map in Hadamard Spaces and Euclidean Buildings

Published online by Cambridge University Press:  20 November 2018

Andreas Balser
Andreas Balser*
Affiliation:
Mathematisches Institut, LMU München, Theresienstrasse 39, D-80333 München, Germany e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that given a stable weighted configuration on the asymptotic boundary of a locally compact Hadamard space, there is a polygon with Gauss map prescribed by the given weighted configuration. Moreover, the same result holds for semistable configurations on arbitrary Euclidean buildings.

Keywords

53C20Euclidean buildingsHadamard spacespolygons
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 3 , 01 September 2006 , pp. 321 - 336
Copyright
Copyright © Canadian Mathematical Society 2006

References

[Bal95] Ballmann, W., Lectures on Spaces of Nonpositive Curvature. DMV Seminar 25, Birkhäuser Verlag, Basel, 1995.Google Scholar
[BH99] Bridson, M. R. and Haefliger, André, Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften 319, Springer-Verlag, Berlin, 1999.Google Scholar
[BL04] Balser, A. and Lytchak, A., Building-like spaces, 2004, arXiv:math.MG/0410437.Google Scholar
[Gau63] Gauss, C. F., Letter to W. Bolyai. Werke, vol. 8, pp. 220–225, 1863. http://gdz.sub.uni-goettingen.de/en/.Google Scholar
[KLM1] Kapovich, M., Leeb, B., and Millson, J. J., Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity. 2004, arXiv: math.DG/0311486.Google Scholar
[KLM2] Kapovich, M., Leeb, B., and Millson, J. J., Polygons in buildings and their refined side lengths. 2004, arXiv:math.MG/0406305.Google Scholar
[KLM3] Kapovich, M., Leeb, B., and Millson, J. J., The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra, to appear in Memoirs of the American Mathematical Society.Google Scholar
[Kar67] Karpelevič, F. I., The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces. Trans. Moscow Math. Soc. 1965(1967), 51–199. American Mathematical Society, Providence, R.I., 1967.Google Scholar
[KL97] Kleiner, B. and Leeb, B., Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Inst. Hautes études Sci. Publ. Math. 86(1997), 115–197.Google Scholar
[Lee97] Leeb, B., A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry. Bonner Mathematische Schritten, Universität Bonn, Mathematisches Institut, 2000.Google Scholar
[Lyt04] Lytchak, A., Rigidity of spherical buildings and joins. To appear in GAFA; http://www.math.uni-bonn.de/people/lytchak/ Google Scholar