Geodesics, distance, and the CAT(0) property for the manifold of Riemannian metrics

Mathematische Zeitschrift

Volumn 273, Issue 1-2, 2013, Pages 55-93

  Clarke, Brian ^a  

^a STANFORD UNIVERSITY   (United States)

Author keywords

CAT(0); Completion; Manifold of metrics; Nonpositive curvature

Indexed keywords

EID: 84872341509     PISSN: 00255874     EISSN: 14321823     Source Type: Journal    
DOI: 10.1007/s00209-012-0996-x     Document Type: Article

References (34)

1. 2 structure of moduli spaces of Einstein metrics on 4-manifolds. Geom. Funct. Anal. 2(1), 29-89 (1992) MR 1143663 (92m: 58017).
2. Besse A. L.: Einstein Manifolds. Classics in Mathematics. Springer, Berlin (2008).
3. Bridson, M. R., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999).
4. Bauer, M., Harms, P., Michor, P. W.: Sobolev metrics on the Riemannian manifold of all Riemannian metrics. Preprint. http://arxiv. org/abs/1102. 3347.
5. Bourguignon J.-P.: Une stratification de l'espace des structures riemanniennes. Compos. Math. 30(1), 1-41 (1975).
6. Coddington E. A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York-Toronto-London (1955).
7. Clarke, B.: The completion of the manifold of Riemannian metrics. J. Differential Geom. (to appear) arXiv: 0904. 0177v1.
8. 2 metric. Ph. D. thesis, University of Leipzig (2009). arXiv: 0904. 0159v1.
9. Clarke B.: The metric geometry of the manifold of Riemannian metrics. Calc. Var. Partial Differ. Equ. 39(3-4), 533-545 (2010) arXiv: 0904. 0174v1.
10. 2 topology on the manifold of Riemannian metrics. Ann. Global Anal. Geom. 39(2), 131-163 (2011).
11. Clarke, B., Rubinstein, Y. A.: Ricci flow and the metric completion of the space of Kähler metrics. Preprint. arXiv: 1102. 3787 (2011).
12. Ebin D. G.: The manifold of Riemannian metrics. Global analysis. In: Chern, S.-S., Smale, S. (eds) Proceedings of Symposia in Pure Mathematics, vol. 15, pp. 11-40. American Mathematical Society, Providence (1970).
13. Freed D. S., Groisser D.: The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group. Michigan Math. J. 36, 323-344 (1989).
14. Gelander T., Karlsson A., Margulis G. A.: Superrigidity, generalized harmonic maps and uniformly convex spaces. Geom. Funct. Anal. 17(5), 1524-1550 (2008).
15. Gil-Medrano, O., Michor, P. W.: The Riemannian manifold of all Riemannian metrics. Q. J. Math. Oxford Ser. (2) 42(166), 183-202 (1991). arXiv: math/9201259.
16. Gromov M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston (2007).
17. Hamilton R. S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. 7(1), 65-222 (1982).
18. Jost J.: Equilibrium maps between metric spaces. Calc. Var. Partial Differ. Equ. 2(2), 173-204 (1994).
19. Jost J.: Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70(4), 659-673 (1995).
20. Jost, J.: Generalized Harmonic Maps between Metric Spaces. Geometric Analysis and the Calculus of Variations, pp. 143-174. International Press, Cambridge (1996).
21. Jost J.: Generalized Dirichlet forms and harmonic maps. Calc. Var. Partial Differ. Equ. 5(1), 1-19 (1997).
22. Jost, J.: Nonpositive curvature: geometric and analytic aspects. Lectures in Mathematics. ETH Zürich, Birkhäuser Verlag, Basel (1997).
23. Klingenberg, W. P. A.: Riemannian Geometry, 2nd edn. de Gruyter Studies in Mathematics, no. 1. Walter de Gruyter, Berlin (1995).
24. Korevaar N. J., Schoen R. M.: Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1(3-4), 561-659 (1993).
25. Korevaar N. J., Schoen R. M.: Global existence theorems for harmonic maps to non-locally compact spaces. Comm. Anal. Geom. 5(2), 333-387 (1997).
26. Lang, S.: Differential and Riemannian Manifolds, 3rd edn. Graduate Texts in Mathematics, vol. 160. Springer, New York (1995).
27. McAlpin, J. H.: Infinite dimensional manifolds and Morse theory. Ph. D. thesis, Columbia University (1965).
28. Michor P. W., Mumford D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217-245 (2005) arXiv: math/0409303.
29. Michor P. W., Mumford D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8(1), 1-48 (2006) arXiv: math. DG/0312384.
30. Palais R. S.: Foundations of Global Non-Linear Analysis. W. A. Benjamin Inc., New York (1968).
31. Rudin W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987).
32. Tromba A. J.: Teichmüller Theory in Riemannian Geometry. Birkhäuser, Basel (1992).
33. Wolpert S. A.: Geodesic length functions and the Nielsen problem. J. Differ. Geom. 25(2), 275-296 (1987).
34. Yamada S.: On the geometry of Weil-Petersson completion of Teichmüller spaces. Math. Res. Lett. 11(2-3), 327-344 (2004).