Skip to main content
SpringerLink
Log in

Harmonic mappings of Riemannian manifolds

  • Conference paper
  • First Online:
Harmonic Mappings and Minimal Immersions

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1161))

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

Access this chapter

Log in via an institution
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 59.95
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.

Bibliography

  1. Baldes, A., Stability and uniqueness properties of the equator map from a ball into an ellipsoid, Math. Z. 185, 505–516 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  2. Baldes, A., Die Äquatorabbildung ins Ellipsoid und harmonische Abbildungen singulärer Riemannscher Mannigfaltigkeiten, Thesis, Bonn 1984.

    Google Scholar 

  3. Benci, V., and J.M. Coron, The Dirichlet problem for harmonic maps from the disk into the Euclidean n-sphere, Preprint.

    Google Scholar 

  4. Brézis, H., and J.M. Coron, Multiple solutions of H-systems and Rellich's conjecture, Comm. Pure Appl. Math. 37, 149–187 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  5. Brézis, H., and J.M. Coron, Large solutions for harmonic maps in two dimensions, Comm. Pure Appl. Math., to appear.

    Google Scholar 

  6. Buser, P., and H. Karcher, Gromov's almost flat manifolds, Astérisque 81 (1981).

    Google Scholar 

  7. Caffarelli, L.A., Regularity theorems for weak solutions of some nonlinear systems, Comm. Pure Appl. Math. 35, 833–838 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  8. Cheng, S.-Y., Liouville theorem for harmonic maps, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc. (1980), 147–151.

    Article  Google Scholar 

  9. Choi, H.J., On the Liouville theorem for harmonic maps, Proc. Amer. Math. Soc. 85, 91–94 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  10. Crittenden, R., Minimum and conjugate points in symmetric spaces, Canad. J. Math. 14, 320–328 (1962).

    Article  MATH  MathSciNet  Google Scholar 

  11. De Giorgi, E., Un esempio di estremali discontinue per un problema variazionale di tipo ellitico, Boll. Un. math. Ital. IV. Ser. 1, 135–138 (1968).

    MATH  Google Scholar 

  12. Ells, J., and J.H. Sampson, Harmonic mappings of Riemannian manifolds, American J. Math. 86, 109–160 (1964).

    Article  Google Scholar 

  13. Eells, J., and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10, 1–68 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  14. Eells, J., and L. Lemaire, Selected topics in harmonic maps, N.S.F. Conference Board Math. Sci., CBMS Regional Confer. Ser. 50 (1983).

    Google Scholar 

  15. Eells, J., and L. Lemaire, Examples of harmonic maps from disks to hemispheres, Math. Z. 185, 517–519 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  16. Fischer-Colbrie, D., Some rigidity theorems for minimal submanifolds of the sphere, Acta Math. 145, 29–46 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  17. Garber, W., S. Ruijsenaars, E. Seiler, and D. Burns, On finite action solutions of the nonlinear σ-model, Ann. Phys. 119, 305–325 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  18. Giaquinta, M., and E. Giusti, On the regularity of the minima of variational integrals, Acta Math. 148, 31–40 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  19. Giaquinta, M., and E. Giusti, The singular set of the minima of quadratic functionals, Ann. Scuola Norm. Sup. Pisa Ser. IV, vol. 11, 45–55 (1984).

    MATH  MathSciNet  Google Scholar 

  20. Giaquinta, M., and S. Hildebrandt, A priori estimates for harmonic mappings, Journal für die Reine u. Angew. Math. 336, 124–164 (1982).

    MathSciNet  Google Scholar 

  21. Giaquinta, M., and J. Souček, Harmonic maps into hemisphere, Preprint.

    Google Scholar 

  22. Gromoll, D., W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Großen, Springer Lecture Notes Nr. 55 (1968).

    Google Scholar 

  23. Grüter, M., and K.-O. Widman, The Green function for uniformly elliptic equations, manuscripta math. 37, 303–342 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  24. Grüter, M., Conformally invariant variational integrals and the removability of isolated singularities, man. math. 47, 85–104 (1984).

    Article  MATH  Google Scholar 

  25. Grüter, M., Regularity of weak H-surfaces, Journ. Reine Angew. Math. 239, 1–15 (1981).

    Google Scholar 

  26. Grüter, M., Regularität stationärer Punkte von konform invarianten Variationsintegralen, Preprint 1984.

    Google Scholar 

  27. Hamilton, R., Harmonic maps of manifolds with boundary, Springer Lecture Notes Nr. 471, 1975.

    Google Scholar 

  28. Hellwig, G., Partielle Differentialgleichungen, Teubner, Stuttgart 1960.

    Google Scholar 

  29. Hildebrandt, S., and H. Kaul, Two-dimensional variational problems with obstructions, and Plateau's problem for H-Surfaces in a Riemannian manifold, Comm. Pure Appl. Math. 25, 187–223 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  30. Hildebrandt, S., and K.-O. Widman, Some regularity results for quasilinear elliptic systems of second order, Math. Z. 142, 67–86 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  31. Hildebrandt, S., and K.-O. Widman, On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order, Ann. Scuola Norm. Sup. Pisa (IV), 4, 145–178 (1977).

    MathSciNet  Google Scholar 

  32. Hildebrandt, S., H. Kaul, K.-O. Widman, Harmonic mappings into Riemannian manifolds with non-positive sectional curvature, Math. Scand. 37, 257–263 (1975).

    MathSciNet  Google Scholar 

  33. Hildebrandt, S., H. Kaul, K.-O. Widman, Dirichlet's boundary value problem for harmonic mappings of Riemannian manifolds, Math. Z. 147, 225–236 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  34. Hildebrandt, S., H. Kaul, and K.-O. Widman, An existence theory for harmonic mappings of Riemannian manifolds, Acta Math. 138, 1–16 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  35. Hildebrandt, S., J. Jost, and K.-O. Widman, Harmonic mappings and minimal surfaces, Inventiones math. 62, 269–298 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  36. Hildebrandt, S., Liouville theorems for harmonic mappings and an approach to Bernstein theorems, Annals of Math. Studies 107–131, Princeton Univ. Press, Seminar on Diff. Geometry, 1982.

    Google Scholar 

  37. Hildebrandt, S., Nonlinear elliptic systems and harmonic mappings, Proc. of the 1980 Beijing Symposium on Diff. Geom. and Diff. Equ., Vol. 1, pp. 481–615, Science Press, Beijing, China 1982.

    MathSciNet  Google Scholar 

  38. Hoffman, D., and R. Osserman, The geometry of the generalized Gauss map, Memoirs of the Amer. Math. Soc., Nov. 1980, Vol. 28, No. 236.

    Google Scholar 

  39. Hoffman, D., and R. Osserman, The area of the generalized Gaussian image and the stability of minimal surfaces in S n and ℝ n, preprint.

    Google Scholar 

  40. Hoffman, D.A., R. Osserman, and R. Schoen, On the Gauss map of complete surfaces of constant mean curvature in ℝ 3 and ℝ 4, Comment. Math. Helv. 57, 519–531 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  41. Ishihara, T., The harmonic Gauss maps in a generalized sense, J. London Math. Soc. 26, 104–112 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  42. Jäger, W., and H. Kaul, Uniqueness and stability of harmonic maps and their Jacobi fields, manuscripta math. 28, 269–291 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  43. Jäger, W., and H. Kaul, Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems, J. Reine Angew. Math. 343, 146–161 (1983).

    MATH  MathSciNet  Google Scholar 

  44. Jost, J., Eine geometrische Bemerkung zu Sätzen über harmonische Abbildungen, die ein Dirichletproblem lösen, manuscripta math. 32, 51–57 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  45. Jost, J., Ein Existenzbeweis für harmonische Abbidungen, die ein Dirichletproblem lösen, mittels der Methode des Wärmeflusses, manuscripta math. 34, 17–25 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  46. Jost, J., A maximum principle for harmonic mappings which solve a Dirichlet problem, manuscripta math. 38, 129–130 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  47. Jost, J., Existence proofs for harmonic mappings with the help of maximum principle, Math. Z. 184, 489–496 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  48. Jost, J., Harmonic maps between surfaces, Springer Lecture Notes Nr. 1062, (1984).

    Google Scholar 

  49. Jost, J., Harmonic mappings between Riemannian manifolds, Proceedings Centre Math. Anal., ANU-Press, Canberra 1984.

    Google Scholar 

  50. Jost, J., and H. Karcher, Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, manuscripta math. 40, 27–77 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  51. Jost, J., and M. Meier, Boundary regularity for minima of certain quadratic functionals, Math. Ann. 262, 549–561 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  52. Karcher, H., Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30, 509–541 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  53. Karcher, H., and J.C. Wood, Non-existence results and growth properties for harmonic maps and forms, SFB-40 preprint, Bonn 1983.

    Google Scholar 

  54. Kasue, A., A gap theorem for minimal submanifolds of Euclidean space, Preprint 1984.

    Google Scholar 

  55. Lawson, H.B., and R. Osserman, Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system, Acta Math. 139, 1–17 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  56. Leichtweiss, K., Zur Riemannschen Geometrie in Grassmannschen Mannigfaltigkeiten, Math. Z. 76, 334–366 (1961).

    Article  MATH  MathSciNet  Google Scholar 

  57. Lemaire, L., Applications harmoniques de surfaces Riemanniennes, Journ. Diff. Geom. 13, 51–78 (1978).

    MATH  MathSciNet  Google Scholar 

  58. Lemaire, L., Minima and critical points of the energy in dimension two, Springer Lecture Notes 838, 187–193 (1981).

    MathSciNet  Google Scholar 

  59. Lemaire, L., Existence des applications harmoniques et courbure des variétés, Séminaire Bourbaki, no. 553 (1979/80).

    Google Scholar 

  60. Littman, W., G. Stampacchia, and H.F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa, Sci. fis. mat. III. Ser. 17, 43–77 (1963).

    MATH  MathSciNet  Google Scholar 

  61. Meier, M., Liouville theorems for nonlinear elliptic equations and systems, manuscripta math. 29, 207–228 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  62. Meier, M., Removable singularities of harmonic maps and an application to minimal submanifolds, to appear in Indiana Math. J.

    Google Scholar 

  63. Meier, M., On quasilinear elliptic systems with quadratic growth, Preprint 1984.

    Google Scholar 

  64. Morrey, C.B., The problem of Plateau on a Riemannian manifold, Ann. of Math. 49, 807–851 (1948).

    Article  MATH  MathSciNet  Google Scholar 

  65. Morrey, S.B., Multiple integrals in the calculus of variations, Springer, Berlin-Heidelberg-New York (1968).

    Google Scholar 

  66. Moser, J., On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14, 577–591 (1961).

    Article  MATH  MathSciNet  Google Scholar 

  67. Osserman, R., Minimal varieties, Bull. Amer. Math. Soc. 75, 1092–1120 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  68. Osserman, R., Minimal surfaces, Gauss maps, total curvature, eigenvalue estimates and stability, The Chern Symposium 1979, Springer 1980.

    Google Scholar 

  69. Price, P., A monotonicity formula for Yang-Mills fields, manuscripta math. 43, 131–166 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  70. Pochozaev, Eigenfunctions of the equation Δu+λf(u)=0, Dokl. Akad. Nauk SSSR 165, 33–36 (1965).

    MathSciNet  Google Scholar 

  71. Ruh, E.A., Asymptotic behaviour of non-parametric minimal hypersurfaces, J. Diff. Geometry 4, 509–513 (1970).

    MATH  MathSciNet  Google Scholar 

  72. Ruh, E.A., and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149, 569–573 (1970).

    Article  MATH  MathSciNet  Google Scholar 

  73. Sacks, J., and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. Math. 113, 1–24 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  74. Sacks, J., and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271, 639–652 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  75. Schoen, R., Analytic aspects of the harmonic map problem, Preprint 1984.

    Google Scholar 

  76. Schoen, R., and K. Uhlenbeck, A regularity theory for harmonic maps, Journal of Diff. Geometry 17, 307–335 (1982); 18, 329 (1983).

    MATH  MathSciNet  Google Scholar 

  77. Schoen, R., and K. Uhlenbeck, Boundary regularity and miscellaneous results on harmonic maps, Journal of Diff. Geometry 18, 253–268 (1983).

    MATH  MathSciNet  Google Scholar 

  78. Schoen, R., and K. Uhlenbeck, Regularity of minimizing harmonic maps into the sphere, Inventiones math., to appear.

    Google Scholar 

  79. Sperner, E. jr., A priori gradient estimates for harmonic mappings, SFB 72-Preprint Nr. 513, Bonn (1982).

    Google Scholar 

  80. Struwe, M., A counterexample in elliptic regularity theory, manuscripta math. 34, 85–92 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  81. Struwe, M., Nonuniqueness in the Plateau problem for surfaces of constant mean curvature, Arch. Rat. Mech. Anal., to appear.

    Google Scholar 

  82. Tachikawa, A., On interior regularity and Liouville's theorem for harmonic mappings, manuscripta math. 42, 11–40 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  83. Tolksdorf, P., A strong maximum principle and regularity for harmonic mappings, SFB 72-Preprint Nr. 586, Bonn 1983.

    Google Scholar 

  84. Tomi, F., Ein einfacher Beweis eines Regularitätssatzes für schwache Lösungen gewisser elliptischer Systeme, Math. Z. 112, 214–218 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  85. Wente, H., The differential equation Δx=2hx u Λx v with vanishing boundary values, Proc. Amer. Math. Soc. 50, 131–137 (1975).

    MATH  MathSciNet  Google Scholar 

  86. Widman, K.-O., The singularity of the Green function for non-uniformly elliptic partial differential equations with discontinuous coefficients, Uppsala University (1970).

    Google Scholar 

  87. Wiegner, M., Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme, Math. Z. 147, 21–28 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  88. Wiegner, M., A-priori Schranken für Lösungen gewisser elliptischer Systeme, manuscripta math. 18, 279–297 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  89. Wiegner, M., Das Existenz-und Regularitätsproblem bei Systemen nichtlinearer elliptischer Differentialgleichungen, Habilitationsschrift, Bochum (1977).

    Google Scholar 

  90. Wiegner, M., On two-dimensional elliptic systems with a one-sided condition, Math. Z. 178, 493–500 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  91. Wolf, J.A., Spaces of constant curvature, New York, Mc Graw-Hill 1966.

    Google Scholar 

  92. Wood, J.C., Non-existence of solutions to certain Dirichlet problems, Preprint 1981.

    Google Scholar 

  93. Wong, Y.-C., Differential geometry of Grassmann manifolds, Proc. Nat. Acad. Sci. USA 57, 589–594 (1967).

    Article  MATH  Google Scholar 

  94. Wong, Y.-C., Sectional curvatures of Grassmann manifolds, Proc. Nat. Acad. Sci. USA 60, 75–79 (1968).

    Article  MATH  Google Scholar 

Download references

Authors
  1. Stefan Hildebrandt
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Istituto Matematico “U.DINI”, Università di Firenze, Viale Morgagni 67/a, 50134, Firenze, Italy

    Enrico Giusti

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Hildebrandt, S. (1985). Harmonic mappings of Riemannian manifolds. In: Giusti, E. (eds) Harmonic Mappings and Minimal Immersions. Lecture Notes in Mathematics, vol 1161. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075136

Download citation

  • DOI: https://doi.org/10.1007/BFb0075136

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16040-3

  • Online ISBN: 978-3-540-39716-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation