Skip to main content
SpringerLink
Log in

The homological singularities of maps in trace spaces between manifolds

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We deal with mappings defined between Riemannian manifolds that belong to a trace space of Sobolev functions. The homological singularities of any such map are represented by a current defined in terms of the boundary of its graph. Under suitable topological assumptions on the domain and target manifolds, we show that the non triviality of the singular current is the only obstruction to the strong density of smooth maps. Moreover, we obtain an upper bound for the minimal integral connection of the singular current that depends on the fractional norm of the mapping.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Almgren, F.J., Browder, W., Lieb, E.H.: Co-area, liquid crystals, and minimal surfaces. In: Partial Differential Equations, Springer Lecture Notes in Math. 1306, 1–22 (1988)

  3. Bethuel F.: A characterization of maps in \({H^1(B^3,\mathbb {S}^2)}\) which can be approximated by smooth maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 7, 269–286 (1990)

    MATH  MathSciNet  Google Scholar 

  4. Bethuel F.: Approximations in trace spaces defined between manifolds. Nonlinear Anal 24, 121–130 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bethuel F., Demengel F.: Extensions for Sobolev mappings between manifolds. Calc. Var. Partial Differ. Equ. 3, 475–491 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bourgain J., Brezis H., Mironescu P.: Lifting in Sobolev spaces. J. Anal. Math. 80, 37–86 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bourgain J., Brezis H., Mironescu P.: On the structure of the Sobolev space H 1/2 with values into the circle. C.R. Acad. Sci. Paris 331, 119–124 (2000)

    MATH  MathSciNet  Google Scholar 

  8. Bourgain J., Brezis H., Mironescu P.: H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg Landau equation. Publ. Math. Inst. Hautes Études Sci. 99, 1–115 (2004)

    MATH  MathSciNet  Google Scholar 

  9. Bourgain J., Brezis H., Mironescu P.: Lifting, degree, and distributional Jacobian revisited. Comm. Pure Appl. Math. 58, 529–551 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bousquet P.: Topological singularities in \({W^{s,p}(\mathbb {S}^N,\mathbb {S}^1)}\). J. Anal. Math. 102, 311–346 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bousquet P.: Fractional Sobolev spaces and topology. Nonlinear Anal. 68, 804–827 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Brezis H., Li Y., Mironescu P., Nirenberg L.: Degree and Sobolev spaces. Topol. Methods Nonlinear Anal. 13, 181–190 (1999)

    MATH  MathSciNet  Google Scholar 

  13. Brezis H., Mironescu P.: On some questions of topology for \({\mathbb {S}^1}\) -valued fractional Sobolev spaces. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95, 121–143 (2001)

    MATH  MathSciNet  Google Scholar 

  14. Brezis H., Nirenberg L.: Degree theory and BMO. I. Compact manifolds without boundaries. Selecta Math. (N.S.) 1, 197–263 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Escobedo M.: Some remarks on the density of regular mappings in Sobolev classes of \({\mathbb {S}^M}\) -valued functions. Rev. Mat. Univ. Complut. Madrid 1, 127–144 (1988)

    MATH  MathSciNet  Google Scholar 

  16. Federer H.: Geometric measure theory. Grundlehren math. wissen. 153. Springer, New York (1969)

    Google Scholar 

  17. Federer H.: Real flat chains, cochains and variational problems. Indiana Univ. Math. J. 24, 351–407 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gagliardo E.: Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova 27, 284–305 (1957)

    MATH  MathSciNet  Google Scholar 

  19. Giaquinta M., Modica G., Mucci D.: The relaxed Dirichlet energy of manifold constrained mappings. Adv. Calc. Var. 1, 1–51 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Giaquinta, M., Modica, G., Souček, J.: Cartesian currents in the calculus of variations, I, II. Ergebnisse Math. Grenzgebiete (III Ser), 37, 38. Springer, Berlin (1998)

  21. Giaquinta M., Modica G., Souček J.: On sequences of maps into \({\mathbb {S}^1}\) with equibounded W 1/2 energies. Selecta Math. (N.S.) 10, 359–375 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Giaquinta M., Mucci D.: Density results for the W 1/2 energy of maps into a manifold. Math. Z. 251, 535–549 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Giaquinta M., Mucci D.: On sequences of maps into a manifold with equibounded W 1/2-energies. J. Funct. Anal. 225, 94–146 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Giaquinta, M., Mucci, D.: Maps into manifolds and currents: area and W 1,2-, W 1/2-, BV-energies. Edizioni della Normale, C.R.M. Series, Sc. Norm. Sup. Pisa (2006)

  25. Hang F., Lin F.: A remark on the Jacobians. Comm. Contemp. Math. 2, 35–46 (2000)

    MATH  MathSciNet  Google Scholar 

  26. Hang F., Lin F.: Topology of Sobolev mappings. II. Acta Math 191, 55–107 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. Hardt R., Lin F.: Mappings minimizing the L p norm of the gradient. Comm. Pure Appl. Math. 40, 555–588 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  28. Hardt, R., Pitts, J.: Solving the Plateau’s problem for hypersurfaces without the compactness theorem for integral currents. In: Allard, W.K., Almgren, F.J. (eds.) Geometric Measure Theory and the Calculus of Variations. Proc. Symp. Pure Math., Am. Math. Soc., Providence, 44, 255–295 (1996)

  29. Hardt R., Rivière T.: Connecting topological Hopf singularities. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 2, 287–344 (2003)

    MATH  Google Scholar 

  30. Hardt R., Rivière T.: Connecting rational homotopy type singularities. Acta Math. 200, 15–83 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Hu S.T.: Homotopy theory. Academic press, New York (1959)

    MATH  Google Scholar 

  32. Isobe T.: Obstructions to the extension problem of Sobolev mappings. Topol. Methods Nonlinear Anal. 21, 345–368 (2003)

    MATH  MathSciNet  Google Scholar 

  33. Kuwert E.: A compactness result for loops with an H 1/2-bound. J. Reine Angew. Math. 505, 1–22 (1998)

    MATH  MathSciNet  Google Scholar 

  34. Millot V., Pisante A.: Relaxed energies for H 1/2-maps with values into the circle and measurable weights. Indiana Univ. Math. J. 58, 49–136 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  35. Mironescu, P.: On some properties of \({\mathbb {S}^1}\) -valued fractional Sobolev spaces. In: Noncompact problems at the intersection of geometry, analysis, and topology, Contemp. Math., 350, Am. Math. Soc., Providence, RI, pp. 201–207 (2004)

  36. Mironescu P., Pisante A.: A variational problem with lack of compactness for \({H^{1/2}(\mathbb {S}^1;\mathbb {S}^1)}\) maps of prescribed degree. J. Funct. Anal. 217, 249–279 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  37. Morgan F.: Area minimizing currents bounded by multiples of curves. Rend. Circ. Mat. Palermo 33, 37–46 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  38. Mucci D.: Strong density results in trace spaces of maps between manifolds. Manuscr. Math. 128(1), 421–441 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  39. Müller T.: Compactness for maps minimizing the n-energy under a free boundary constraint. Manuscr. Math. 103, 513–540 (2000)

    Article  MATH  Google Scholar 

  40. Pakzad M.R., Rivière T.: Weak density of smooth maps for the Dirichlet energy between manifolds. Geom. Funct. Anal. 13, 223–257 (2001)

    Article  Google Scholar 

  41. Rivière T.: Dense subsets of \({H^{1/2}(\mathbb {S}^2;\mathbb {S}^1)}\). Ann. Glob. Anal. Geom. 18, 517–528 (2000)

    Article  MATH  Google Scholar 

  42. Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Math. Analysis, vol. 3 Australian National University, Canberra (1983)

  43. White B.: The least area bounded by multiples of a curve. Proc. Am. Math. Soc. 90, 230–232 (1984)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica dell’Università di Parma, Viale G. P. Usberti 53/A, 43100, Parma, Italy

    Domenico Mucci

Authors
  1. Domenico Mucci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Domenico Mucci.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mucci, D. The homological singularities of maps in trace spaces between manifolds. Math. Z. 266, 817–849 (2010). https://doi.org/10.1007/s00209-009-0600-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-009-0600-1

Keywords

Mathematics Subject Classification (2000)

Search

Navigation