Mathematische Annalen

, Volume 373, Issue 1–2, pp 155–163 | Cite as

Conformal deformations of CAT(0) spaces

  • Alexander LytchakEmail author
  • Stephan Stadler


We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.

Mathematics Subject Classification

53C20 53C23 58E20 



The authors are grateful to Stefan Wenger for very helpful comments. Both authors were partially supported by DFG grant SPP 2026.


  1. 1.
    Alexander, S., Bishop, R.: Curvature bounds for warped products of metric spaces. GAFA 14, 1143–1181 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alexander, S., Kapovitch, V., Petrunin, A.: Alexandrov geometry. (2017)
  3. 3.
    Ballmann, W.: On the geometry of metric spaces. Lecture notes, (2004)
  4. 4.
    Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence, RI (2001)Google Scholar
  5. 5.
    Evans, Lawrence C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)Google Scholar
  6. 6.
    Fuglede, B.: The Dirichlet problem for harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature. Trans. Am. Math. Soc. 357(2), 757–792 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Guo, C., Wenger, S.: Area minimizing discs in locally non-compact metric spaces. Comm. Anal. Geom., to appear. Preprint arXiv:1701.06736 (2017)
  8. 8.
    Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.: Sobolev spaces on metric measure spaces. Cambridge University Press (2015)Google Scholar
  9. 9.
    Jost, J.: Equilibrium maps between metric spaces. Calc. Var. Partial Differ. Equ. 2(2), 173–204 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Korevaar, N., Schoen, R.: Sobolev spaces and harmonic maps for metric space targets. Commun. Anal. Geom. 1(3–4), 561–659 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lytchak, A., Wenger, S.: Intrinsic structure of minimal discs in metric spaces. Geom. Topol. 22(1), 591–644 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lytchak, A., Wenger, S.: Isoperimetric characterization of upper curvature bounds. Preprint arXiv:1611.05261 (2016)
  13. 13.
    Lytchak, A., Wenger, S.: Area minimizing discs in metric spaces. Arch. Ration. Mech. Anal. 223(3), 1123–1182 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mese, C.: The curvature of minimal surfaces in singular spaces. Commun. Anal. Geom. 9, 1–34 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Petrunin, A.: Semiconcave functions in Alexandrov’s geometry. Surveys in Differential Geometry, no. 92, 11, Int. Press, Somerville, MA, 137–201 (2007)Google Scholar
  16. 16.
    Petrunin, A., Stadler, S.: Metric minimizing surfaces revisited. Preprint arXiv:1707.09635 (2017)
  17. 17.
    Reshetnyak, Y.G.: Two-dimensional manifolds of bounded curvature. In Geometry, IV, volume 70 of Encyclopaedia Math. Sci., pp. 3–163, 245-250. Springer, Berlin (1993)Google Scholar
  18. 18.
    Sturm, K.-T.: Ricci tensor for diffusion operators and curvature-dimension inequalities under conformal transformations and time changes. J. Funct. Anal. 275(4), 793–829 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität KölnKölnGermany
  2. 2.Mathematisches Institut der Universität MünchenMünchenGermany

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