The local invariant for scale structures on mapping spaces

  • Jungsoo KangEmail author


A scale Hilbert space is a natural generalization of a Hilbert space which considers not only a single Hilbert space but a nested sequence of subspaces. Scale structures were introduced by H. Hofer, K. Wysocki, and E. Zehnder as a new concept of smooth structures in infinite dimensions. In this paper, we prove that scale structures on mapping spaces are completely determined by the dimension of domain manifolds. We also give a complete description of the local invariant introduced by U. Frauenfelder for these spaces. Product mapping spaces and relative mapping spaces are also studied. Our approach is based on the spectral resolution of Laplace type operators together with the eigenvalue growth estimate.


Scale structures Frauenfelder’s local invariant Mapping spaces 

Mathematics Subject Classification

46T05 58B15 



I am grateful to Urs Frauenfelder for fruitful discussions. I thank Jeong Hyeong Park for precious communications. I also thank the referee for careful reading of the manuscript and helpful comments.


  1. 1.
    Bérard, P.H.: Spectral Geometry: Direct and Inverse Problems. Lecture Notes in Mathematics. Springer, Berlin (2006)Google Scholar
  2. 2.
    Chavel, I.: Eigenvalues in Riemannian Geometry. Academic Press, London (1994)zbMATHGoogle Scholar
  3. 3.
    Dodziuk, J.: Sobolev spaces of differential forms and deRham–Hodge isomorphism. J. Differ. Geom. 16, 63–73 (1981)CrossRefGoogle Scholar
  4. 4.
    Eliasson, H.: Geometry of manifolds of maps. J. Differ. Geom. 1, 169–194 (1967)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Floer, A.: Morse theory for Lagrangian intersections. J. Differ. Geom. 28, 513–547 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Frauenfelder, U.: First steps in the geography of scale Hilbert structures (2009), arXiv:0910.3980
  7. 7.
    Frauenfelder, U.: Fractal scale Hilbert spaces and scale Hessian operators (2009), arXiv:0912.1154
  8. 8.
    Gilkey, P.: Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1995)zbMATHGoogle Scholar
  9. 9.
    Gilkey, P.: The Spectral Geometry of Operators of Dirac and Laplace Type, “Handbook of Global Analysis”. Elsevier, Amsterdam (2008)zbMATHGoogle Scholar
  10. 10.
    Gilkey, P., Leahy, J., Park, J.H.: Spinors, Spectral Geometry, and Riemannian Submersions. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center (1998)Google Scholar
  11. 11.
    Grubb, G.: Functional Calculus of Pseudodifferential Boundary Problems. Progress in Mathematics, vol. 65. Birkhäuser, Boston (1996)CrossRefGoogle Scholar
  12. 12.
    Hohloch, S., Noetzel, G., Salamon, D.: Hypercontact structures and Floer theory. Geom. Topol. 13, 2543–2617 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hofer, H., Wysocki, K., Zehnder, E.: Fredholm Theory in Polyfolds I: Functional Analytic Methods (Book in preparation) Google Scholar
  14. 14.
    Hofer, H., Wysocki, K., Zehnder, E.: A general Fredholm theory I: a splicing-based differential geometry. J. Eur. Math. Soc. 9, 841–876 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jost, J.: Partial Differential Equations. Springer, Berlin (2002)zbMATHGoogle Scholar
  16. 16.
    Jost, J.: Postmodern Analysis. Springer, Berlin (2005)zbMATHGoogle Scholar
  17. 17.
    Klingenberg, W.: Lectures on Closed Geodesics. Springer, Berlin (1978)CrossRefGoogle Scholar
  18. 18.
    McDuff, D., Salamon, D.: J-Holomorphic Curves and Symplectic Topology, vol. 52. American Mathematical Society Colloquium Publications, Providence (2004)zbMATHGoogle Scholar
  19. 19.
    Oh, Y.-G.: Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks I. Comm. Pure Appl. Math. 46, 949–994 (1993)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Salomonsen, G.: Equivalence of Sobolev spaces. Results Math. 39, 115–130 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wehrheim, K.: Uhlenbeck Compactness, EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich (2004) Google Scholar

Copyright information

© The Author(s), under exclusive licence to Mathematisches Seminar der Universität Hamburg 2019

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, Research Institute in MathematicsSeoul National UniversityGwanak-Gu, SeoulSouth Korea

Personalised recommendations