Skip to main content

Open problems in the degree theory for disc minimal surfaces spanning a curve in ℝ3

  • Chapter
  • First Online:
Partial Differential Equations and Calculus of Variations

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1357))

  • 1681 Accesses

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alt, H.W.: Verzweigungspunkte von H-Flächen I, Math. Z. 127, 333–362 (1972); II, Math. Ann. 201, 33–56 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  2. Böhme, R. and Tromba, A.J.: The index theorem for classical minimal surfaces; Annals of Mathematics 113 (1981), pp. 447–499

    Article  MathSciNet  MATH  Google Scholar 

  3. Elworthy, K.D. and Tromba, A.J.: Differential structures and Fredholm maps on Banach manifolds; Proc. Pure Math. vol. 15, AMS (1970), pp. 45–94.

    Article  MathSciNet  MATH  Google Scholar 

  4. Elworthy, K.D. and Tromba, A.J.: Degree theory on Banach manifolds; Proc. Symp. Pure Math. Vol 18, AMS (1970), pp. 86–94.

    Article  MathSciNet  MATH  Google Scholar 

  5. Gulliver, R.: Regularity of minimizing surfaces of prescribed mean curvature; Ann. of Math. 97, 275–305 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  6. Heinz, E. and Tomi, F.: Zu einem Satz von Hildebrandt über das Randverhalten von Minimalflächen; Math. Z. 111 (1969), 372–386.

    Article  MathSciNet  MATH  Google Scholar 

  7. Hildebrandt, S.: Boundary behavior of minimal surfaces; Arch. Rational Mech. Anal. 35 (1969), pp. 47–81.

    Article  MathSciNet  MATH  Google Scholar 

  8. Leray, J. and Schauder, J.: Topologie et equations fonctionelles; Ann. Sci. Ecole Norm. Sup. 51 (1934), pp. 45–78.

    MathSciNet  MATH  Google Scholar 

  9. Nirenberg, L.: Variational and topological methods in non-linear problems; Bull. AMS 4 (1981), 267–302.

    Article  MathSciNet  MATH  Google Scholar 

  10. Osserman, R.: A proof of the regularity everywhere of the classical solution of Plateau’s problem; Ann. of Math. (2) 91 (1970), pp. 550–569.

    Article  MathSciNet  MATH  Google Scholar 

  11. Smale, S.: An infinite dimensional version of Sard’s theorem; Amer. J. Math. 87 (1965), pp. 861–866.

    Article  MathSciNet  MATH  Google Scholar 

  12. Struwe, M.: On a critical point theory of minimal surfaces spanning a wire in IRn; J. Reine Angew. Math. 349 (1984), pp. 1–23.

    MathSciNet  MATH  Google Scholar 

  13. Tomi, F. and Tromba, A.J.: On the structure of the set of curves bounding minimal surfaces of prescribed degeneracy; J. Reine Angew. Math. 316 (1980), 31–43.

    MathSciNet  MATH  Google Scholar 

  14. Thiel, U.: On the stratification of branched minimal surfaces; Analysis 5 (1985), 251–271.

    Article  MathSciNet  MATH  Google Scholar 

  15. Tromba, A.J.: Degree theory on oriented infinite dimensional varieties and the Morse number spanning a curve in IRn, Part II, n = 3; Manuscripta math. 48 (1984), pp. 139–161; Part I, Trans. AMS, vol. 290, Number 1 (July 1985), pp. 385–413.

    Article  MathSciNet  MATH  Google Scholar 

  16. Tromba, A.J.: A general asymptotic fixed point theorem; J. Reine Angew. Math. 332 (1982), pp. 118–123.

    MathSciNet  MATH  Google Scholar 

  17. Tromba, A.J.: On the number of simply connected minimal surfaces spanning a curve; Memoirs of the AMS, 12 194, (1977).

    Google Scholar 

Download references

Authors

Editor information

Stefan Hildebrandt Rolf Leis

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag

About this chapter

Cite this chapter

Tromba, A.J. (1988). Open problems in the degree theory for disc minimal surfaces spanning a curve in ℝ3 . In: Hildebrandt, S., Leis, R. (eds) Partial Differential Equations and Calculus of Variations. Lecture Notes in Mathematics, vol 1357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0082876

Download citation

  • DOI: https://doi.org/10.1007/BFb0082876

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50508-2

  • Online ISBN: 978-3-540-46024-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics