Skip to main content
SpringerLink
Log in
Book cover

Coulomb Frames in the Normal Bundle of Surfaces in Euclidean Spaces pp 53–73Cite as

Normal Coulomb Frames in \({\mathbb{R}}^{4}\)

  • Chapter
  • First Online:

  • 795 Accesses

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2053))

Abstract

With this chapter we begin our study of constructing normal Coulomb frames, here for surfaces immersed in Euclidean space \({\mathbb{R}}^{4}.\)Normal Coulomb frames are critical for a new functional of total torsion. We present the associated Euler–Lagrange equation and discuss its solution via a Neumann boundary value problem. A proof of the “minimal character” of normal Coulomb frames follows immediately.Using methods from potential theory and complex analysis we establish various analytical tools to control these special frames. For example, we present two different methods to bound their torsion (connection) coefficients. Methods from the theory of generalized analytic functions will play again an important role.We conclude the third chapter with a class of minimal graphs for which we can explicitly compute normal Coulomb frames.

This is a preview of subscription content, 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   34.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.95
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

Learn about institutional subscriptions

Notes

  1. 1.

    In Sect. 2.1.2 we have already discussed surfaces with parallel mean curvature vector and a possible connection to the curvatures of their normal bundles.

  2. 2.

    The constants c 1, c 2 and C 1, C 2 below depend also on the domain \(\overline{B}.\)

References

  1. G. Albrecht, The Veronese surface revisited. J. Geom. 73, 22–38 (2002)

    Google Scholar 

  2. W. Alt, Lineare Funktionalanalysis (Springer, New York, 2006)

    Google Scholar 

  3. B. Andrews, C. Baker, Mean curvature flow of pinched submanifolds to spheres. J. Differ. Geom. 85(3), 357–396 (2010)

    Google Scholar 

  4. C. Bär, Elementare Differentialgeometrie (Walter de Gruyter GmbH & Co. KG, Berlin, 2010)

    Google Scholar 

  5. J.L. Barbosa, M. do Carmo, On the size of stable minimal surfaces in \({\mathbb{R}}^{3}\). Am. J. Math. 98, 515–528 (1974)

    Google Scholar 

  6. J.L. Barbosa, M. do Carmo, Stability of minimal surfaces and eigenvalues of the Laplacian. Math. Z. 173, 13–28 (1980)

    Google Scholar 

  7. H.W. Begehr, Complex Analytic Methods for Partial Differential Equations (World Scientific Publishing, River Edge, 1994)

    Google Scholar 

  8. M. Bergner, S. Fröhlich, On two-dimensional immersions of prescribed mean curvature in \({\mathbb{R}}^{n}\). Z. Anal. Anw. 27(1), 31–52 (2008)

    Google Scholar 

  9. M. Bergner, R. Jakob, Exclusion of boundary branch points for minimal surfaces. Analysis 31, 181–190 (2011)

    Google Scholar 

  10. S. Bernstein, Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus. Math. Z. 26, 551–558 (1927)

    Google Scholar 

  11. L. Bers, Univalent solutions of linear elliptic systems. Comm. Pure Appl. Math. VI, 513–526 (1953)

    Google Scholar 

  12. W. Blaschke, K.Leichtweiss, Elementare Differentialgeometrie (Springer, New York, 1973)

    Google Scholar 

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

    Google Scholar 

  14. H. Brauner, Differentialgeometrie (Friedrich Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig, 1981)

    Google Scholar 

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

    Google Scholar 

  16. R. Bryant, A duality theorem for Willmore surfaces. J. Differ. Geom. 20, 23–53 (1984)

    Google Scholar 

  17. J.L. Buchanan, A similarity principle for Pascali systems. Complex Variables 1, 155–165 (1983)

    Google Scholar 

  18. J.L. Buchanan, R.P. Gilbert, First Order Elliptic Systems (Academic, New York, 1983)

    Google Scholar 

  19. A. Burchard, L.E. Thomas, On the Cauchy problem for a dynamical Euler’s elastica. Comm. Part. Differ. Equat. 28, 271–300 (2003)

    Google Scholar 

  20. H. Cartan, Differential forms (Dover publications, Mineola, New York, 2006)

    Google Scholar 

  21. B.Y. Chen, Geometry of Submanifolds (Marcel Dekker, New York, 1973)

    Google Scholar 

  22. B.Y. Chen, in Riemannian Submanifolds. Handbook of Differential Geometrie 1 (Elsevier Science B.V., The Netherlands, 1999), pp. 187–418

    Google Scholar 

  23. B.Y. Chen, G.D. Ludden, Surfaces with mean curvature vector parallel in the normal bundle. Nagoya Math. J. 47, 161–167 (1972)

    Google Scholar 

  24. S.S. Chern, An elementary proof of the existence of isothermal parameters on a surface. Proc. Am. Math. Soc. 6(5), 771–782 (1955)

    Google Scholar 

  25. M.A. Cheshkova, Evolute surfaces in E 4. Math. Notes 70(6), 870–872 (2001)

    Google Scholar 

  26. R. Coifman, P.L. Lions, Y. Meyer, S. Semmes, Compensated compactness and Hardy spaces. J. Math. Pures Appl. 9(3), 247–286 (1993)

    Google Scholar 

  27. T.H. Colding, W.P. Minicozzi, Minimal Surfaces (Courant Institute of Mathematical Sciences, New York, 1999)

    Google Scholar 

  28. R. Courant, Dirichlet’s Principle, Conformal Mappings, and Minimal Surfaces (Interscience publishing, New York, 1950)

    Google Scholar 

  29. R. Courant, D. Hilbert, Methods of Mathematical Physics 2 (Wiley, New York, 1962)

    Google Scholar 

  30. R.C.T. da Costa, Constraints in quantum mechanics. Phys. Rev. A25(6), 2893–2900 (1982)

    Google Scholar 

  31. A. Dall’Acqua, Uniqueness for the homogeneous Dirichlet Willmore boundary value problem (2012). Ann. Glob. Anal. Geom. DOI: 10.1007/S10455-012-9320-6

    Google Scholar 

  32. A. Dall’Acqua, K. Deckelnick, H.-Chr. Grunau, Classical solutions to the Dirichlet problem for Willmore surfaces of revolution. Adv. Calc. Var. 1, 379–397 (2008)

    Google Scholar 

  33. U. Dierkes, Maximum principles for submanifolds of arbitrary codimension and bounded mean curvature. Calc. Var. 22, 173–184 (2005)

    Google Scholar 

  34. U. Dierkes, S. Hildebrandt, F. Sauvigny, Minimal Surfaces (Springer, New York, 2010)

    Google Scholar 

  35. P.M. Dirac, Lectures on Quantum Mechanics (Dover Publications, New York, 2001)

    Google Scholar 

  36. M.P. do Carmo, Riemannian Geometry (Birkhäuser, Boston, 1992)

    Google Scholar 

  37. M. Dobrowolski, Angewandte Funktionalanalysis (Springer, New York, 2006)

    Google Scholar 

  38. H. Dorn, G. Jorjadze, S. Wuttke, On spacelike and timelike minimal surfaces in AdSn. JHEP 05 (2009) 048. DOI: 10.1088/1126-6708/2009/05/048

    Google Scholar 

  39. P. Duren, Theory of H p Spaces (Dover Publications, New York, 2000)

    Google Scholar 

  40. P. Duren, Harmonic Mappings in the Plane (Cambridge University Press, Cambridge, 2004)

    Google Scholar 

  41. K. Ecker, Regularity Theory for Mean Curvature Flow (Birkhäuser, Boston, 2004)

    Google Scholar 

  42. K. Ecker, G. Huisken, Interior curvature estimates for hypersurfaces of prescribed mean curvature. Ann. Inst. H. Poincaré Anal. Non Linéaire 6, 251–260 (1989)

    Google Scholar 

  43. L.P. Eisenhart, Riemannian Geometry (Princeton University Press, Princeton, 1949)

    Google Scholar 

  44. J.-H. Eschenburg, J. Jost, Differentialgeometrie und Minimalflächen (Springer, New York, 2007)

    Google Scholar 

  45. C. Fefferman, E.M. Stein, H p spaces of several variables. Acta Math. 129, 137–193 (1972)

    Google Scholar 

  46. S. Fröhlich, Curvature estimates for μ-stable G-minimal surfaces and theorems of Bernstein type. Analysis 22, 109–130 (2002)

    Google Scholar 

  47. S. Fröhlich, Katenoidähnliche Lösungen geometrischer Variationsprobleme (2004). Preprint 2322, FB Mathematik, TU Darmstadt

    Google Scholar 

  48. S. Fröhlich, On 2-surfaces in \({\mathbb{R}}^{4}\) and \({\mathbb{R}}^{n}\). in Proceedings of the 5th Conference of Balkan Society of Geometers, Mangalia, 2005

    Google Scholar 

  49. S. Fröhlich, F. Müller, On critical normal sections for two-dimensional immersions in \({\mathbb{R}}^{4}\) and a Riemann–Hilbert problem. Differ. Geom. Appl. 26, 508–513 (2008)

    Google Scholar 

  50. S. Fröhlich, F. Müller, On critical normal sections for two-dimensional immersions in \({\mathbb{R}}^{n+2}\). Calc. Var. 35, 497–515 (2009)

    Google Scholar 

  51. S. Fröhlich, F. Müller, On the existence of normal Coulomb frames for two-dimensional immersions with higher codimension. Analysis 31, 221–236 (2011)

    Google Scholar 

  52. S. Fröhlich, S. Winklmann, Curvature estimates for graphs with prescribed mean curvature and flat normal bundle. Manuscripta Math. 122(2), 149–162 (2007)

    Google Scholar 

  53. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, New York, 1983)

    Google Scholar 

  54. K. Große-Brauckmann, R. Kusner, J. Sullivan, Coplanar constant mean curvature surfaces. Comm. Anal. Geom. 5, 985–1023 (2007)

    Google Scholar 

  55. I.V. Guadalupe, L. Rodriguez, Normal curvature of surfaces in space forms. Pac. J. Math. 106(1), 95–103 (1983)

    Google Scholar 

  56. R. Gulliver, in Minimal Surfaces of Finite Index in Manifolds of Positive Scalar Curvature, ed. by S. Hildebrandt, D. Kinderlehrer, M. Miranda. Calculus of Variations and Partial Differential Equations (Springer, New York, 1988)

    Google Scholar 

  57. R.R. Hall, On an inequality of E. Heinz. J. Anal. Math. 42, 185–198 (1982)

    Google Scholar 

  58. P. Hartman, A. Wintner, On the existence of Riemannian manifolds which cannot carry non-constant analytic or harmonic functions in the small. Am. J. Math. 75, 260–276 (1953)

    Google Scholar 

  59. P. Hartman, A. Wintner, On the local behaviour of solutions of nonparabolic partial differential equations. Am. J. Math. 75, 449–476 (1953)

    Google Scholar 

  60. E. Heil, Differentialformen und Anwendungen auf Vektoranalysis, Differentialgleichungen, Geometrie (Bibliographisches Institut, Germany, 1974)

    Google Scholar 

  61. E. Heinz, Über die Lösungen der Minimalflächengleichung. Nachr. Akad. Gött., Math.-Phys. Kl., 51–56 (1952)

    Google Scholar 

  62. E. Heinz, On certain nonlinear elliptic differential equations and univalent mappings. J. Anal. Math. 5, 197–272 (1957)

    Google Scholar 

  63. E. Heinz, Über das Randverhalten quasilinearer elliptischer Systeme mit isothermen Parametern. Math. Z. 113, 99–105 (1970)

    Google Scholar 

  64. F. Helein, Harmonic Maps, Conservation Laws and Moving Frames (Cambridge University Press, Cambridge, 2002)

    Google Scholar 

  65. W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments. Z. Naturforsch. 28c, 693–703 (1973)

    Google Scholar 

  66. S. Hildebrandt, Einige Bemerkungen über Flächen beschränkter mittlerer Krümmung. Math. Z. 115, 169–178 (1970)

    Google Scholar 

  67. S. Hildebrandt, H. von der Mosel, On Lichtenstein’s theorem about globally conformal mappings. Calc. Var. 23, 415–424 (2005)

    Google Scholar 

  68. S. Hildebrandt, J. Jost, K.O. Widman, Harmonic mappings and minimal submanifolds. Invent. Math. 62, 269–298 (1980)

    Google Scholar 

  69. D. Hoffman, R. Osserman, The Geometry of the Generalized Gauss Map (American Mathematical Society, Providence, 1980)

    Google Scholar 

  70. E. Hopf, Bemerkungen zu einem Satze von S. Bernstein aus der Theorie der elliptischen Differentialgleichungen. Math. Z. 29, 744–745 (1929)

    Google Scholar 

  71. E. Hopf, On S. Bernstein’s theorem on surfaces z(x, y) of nonpositive curvature. Proc. Am. Math. Soc. 1(1), 80–85 (1950)

    Google Scholar 

  72. H. Hopf, Über Flächen mit einer Relation zwischen den Hauptkrümmungen. Math. Nachr. 4, 232–249 (1950/51)

    Google Scholar 

  73. R. Jakob, Finiteness of the set of solutions of Plateau’s problem for polygonal boundary curves. Ann. I.H. Poincaré Anal. Non Linéaire 24, 963–987 (2007)

    Google Scholar 

  74. R. Jakob, About the finiteness of the set of solutions of Plateau’s problem for polygonal boundary curves. Analysis 29, 365–385 (2009)

    Google Scholar 

  75. R. Jakob, Finiteness of the number of solutions of Plateau’s problem for polygonal boundary curves II. Ann. Global Anal. Geom. 36, 19–35 (2009)

    Google Scholar 

  76. J. Jost, Y.L. Xin, Bernstein type theorems for higher codimension. Calc. Var. 9, 277–296 (1999)

    Google Scholar 

  77. H. Kalf, T. Kriecherbauer, E. Wienholtz, Elliptische Differentialgleichungen Zweiter Ordnung (Springer, New York, 2009)

    Google Scholar 

  78. K. Kenmotsu, Surfaces of Constant Mean Curvature (Oxford University Press, Oxford, 2003)

    Google Scholar 

  79. K. Kenmotsu, D. Zhou, The classification of the surfaces with parallel mean curvature vector in two-dimensional complex space forms. Am. J. Math. 122, 295–317 (2000)

    Google Scholar 

  80. W. Klingenberg, Eine Vorlesung über Differentialgeometrie (Springer, New York, 1973)

    Google Scholar 

  81. K. Kobayashi, Fundamental equations for submanifolds. Fortschr. Phys. 37, 599–610 (1989)

    Google Scholar 

  82. B.G. Konopelchenko, G. Landolfi, On rigid string instantons in four dimensions. Phys. Lett. B459, 522–526 (1999)

    Google Scholar 

  83. A. Korn, in Zwei Anwendungen der Methode der sukzessiven Approximation. Mathematische Abhandlungen Hermann Amandus Schwarz zu seinem fünfzigjährigem Doktorjubiläum am 6. August 1914 gewidmet von Freunden und Schülern (Springer, New York, 1914), pp. 215–229

    Google Scholar 

  84. W. Kühnel, Differentialgeometrie (Friedrich Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig, 1999)

    Google Scholar 

  85. H.B. Lawson, Complete minimal surfaces in S 3. Ann. Math. 92, 335–374 (1970)

    Google Scholar 

  86. H. Li, Willmore surfaces in S n. Ann. Glob. Anal. Geom. 21, 203–213 (2002)

    Google Scholar 

  87. L. Lichtenstein, Zur Theorie der konformen Abbildung. Konforme Abbildung nichtanalytischer, singularitätenfreier Flächenstücke auf ebene Gebiete. Bull. Int. de l’Acad. Sci. Cracovie A, 192–217 (1916)

    Google Scholar 

  88. M.J. Micallef, Stable minimal surfaces in Euclidean space. J. Differ. Geom. 19, 57–84 (1984)

    Google Scholar 

  89. R. Moser, Partial Regularity, Harmonic Maps and Related Problems (World Scientific Publishing, River Edge, 2005)

    Google Scholar 

  90. F. Müller, Funktionentheorie und Minimalflächen. Lecture notes, University Duisburg (2011)

    Google Scholar 

  91. F. Müller, A. Schikorra, Boundary regularity via Uhlenbeck-Rivière decomposition. Analysis 29(2), 199–220 (2009)

    Google Scholar 

  92. J.C.C. Nitsche, Vorlesungen über Minimalflächen (Springer, New York, 1975)

    Google Scholar 

  93. R. Osserman, Global properties of minimal surfaces in \({\mathbb{R}}^{3}\) and \({\mathbb{R}}^{n}\). Ann. Math. 80(2), 340–364 (1964)

    Google Scholar 

  94. R. Osserman, A Survey of Minimal Surfaces (Dover Publications, New York, 1986)

    Google Scholar 

  95. B. Palmer, Uniqueness theorems for Willmore surfaces with fixed and free boundaries. Indiana Univ. Math. J. 49, 1581–1601 (2000)

    Google Scholar 

  96. M. Pinl, Abwickelbare Schiebflächen in R n . Comm. Math. Helv. 24, 64–67 (1950)

    Google Scholar 

  97. M. Pinl, B-Kugelbilder reeller Minimalflächen in R 4. Math. Z. 59, 290–295 (1953)

    Google Scholar 

  98. T. Rivière, Conservation laws for conformally invariant variational problems. Invent. Math. 168, 1–22 (2007)

    Google Scholar 

  99. T. Rivière, Analysis aspects of Willmore surfaces. Invent. math. 174, 1–45 (2008)

    Google Scholar 

  100. H. Ruchert, Ein Eindeutigkeitssatz für Flächen konstanter mittlerer Krümmung. Arch. math. 33, 91–104 (1979)

    Google Scholar 

  101. K. Sakamoto, Variational problems of normal curvature tensor and concircular scalar fields. Tohoku Math. J. 55, 207–254 (2003)

    Google Scholar 

  102. F. Sauvigny, Die zweite Variation von Minimalflächen im \({\mathbb{R}}^{p}\) mit polygonalem Rand. Math. Z. 189, 167–184 (1985)

    Google Scholar 

  103. F. Sauvigny, Ein Eindeutigkeitssatz für Minimalflächen im \({\mathbb{R}}^{p}\) mit polygonalem Rand. J. Reine Angew. Math. 358, 92–96 (1985)

    Google Scholar 

  104. F. Sauvigny, On the Morse index of minimal surfaces in ℝ p with polygonal boundaries. Manuscripta Math. 53, 167–197 (1985)

    Google Scholar 

  105. F. Sauvigny, A-priori-Abschätzungen der Hauptkrümmungen für Immersionen vom Mittleren-Krümmungs-Typ mittels Uniformisierung und Sätze vom Bernstein-Typ. Habilitationsschrift, Göttingen (1988)

    Google Scholar 

  106. F. Sauvigny, On immersions of constant mean curvature: Compactness results and finiteness theorems for Plateau’s problem. Arch. Rat. Mech. Anal. 110, 125–140 (1990)

    Google Scholar 

  107. F. Sauvigny, Partielle Differentialgleichungen der Geometrie und Physik (Springer, New York, 2005)

    Google Scholar 

  108. R. Schätzle, The Willmore boundary value problem. Calc. Var. 37, 275–302 (2010)

    Google Scholar 

  109. R. Schoen, L. Simon, S.T. Yau, Curvature estimates for minimal hypersurfaces. Acta Math. 134, 275–288 (1975)

    Google Scholar 

  110. J.A. Schouten, Ricci Calculus (Springer, New York, 1954)

    Google Scholar 

  111. F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampère Equations in Two Dimensions (Springer, New York, 1991)

    Google Scholar 

  112. B. Sharp, P. Topping, Decay estimates for Rivière’s equation, with applications to regularity and compactness. Trans. Am. Math. Soc. (to appear)

    Google Scholar 

  113. K. Smoczyk, G. Wang, Y.L. Xin, Bernstein type theorems with flat normal bundle. Calc. Var. 26, 57–67 (2006)

    Google Scholar 

  114. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (Princeton University Press, Princeton, 1993)

    Google Scholar 

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

    Google Scholar 

  116. P. Topping, The optimal constant in Wente’s L ∞-estimate. Comment. Math. Helv. 72, 316–328 (1997)

    Google Scholar 

  117. I.N. Vekua, Verallgemeinerte Analytische Funktionen (Akademie, Berlin, 1963)

    Google Scholar 

  118. A. Voss, Zur Theorie der Transformation quadratischer Differentialausdrücke und der Krümmung höherer Mannigfaltigkeiten. Math. Ann. 16, 129–179 (1880)

    Google Scholar 

  119. M.-T. Wang, On graphic Bernstein type results in higher codimension. Trans. Am. Math. Soc. 355, 265–271 (2003)

    Google Scholar 

  120. J.L. Weiner, On a problem of Chen, Willmore, et al. Indiana Univ. Math. J. 27(1), 19–35 (1978)

    Google Scholar 

  121. W.L. Wendland, Elliptic Systems in the Plane (Pitman Publishing, London, 1979)

    Google Scholar 

  122. H.C. Wente, The differential equation \(\bigtriangleup x = 2H{x}_{u} \wedge {x}_{v}\) with vanishing boundary values. Proc. Am. Math. Soc. 50, 131–137 (1975)

    Google Scholar 

  123. H.C. Wente, Large solutions to the volume constrained Plateau problem. Arch. Rat. Mech. Anal. 75, 59–77 (1980)

    Google Scholar 

  124. H. Weyl, Zur Infinitesimalgeometrie: p-dimensionale Fläche im n-dimensionalen Raum. Math. Z. 12, 154–160 (1922)

    Google Scholar 

  125. T.J. Willmore, Riemannian Geometry (Oxford Science Publications, Clarendon Press, Oxford, 1993)

    Google Scholar 

  126. W. Wirtinger, Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde und Hermitesche Massbestimmung. Monatsh. Math. Phys. 44, 343–365 (1936)

    Google Scholar 

  127. Y.L. Xin, Curvature estimates for submanifolds with prescribed Gauss image and mean curvature. Calc. Var. 37, 385–405 (2010)

    Google Scholar 

  128. B. Zwiebach, A First Course in String Theory (Cambridge University Press, Cambridge, 2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Mathematics, Computer Sciences, Johannes Gutenberg University, Mainz, Germany

    Steffen Fröhlich

Authors
  1. Steffen Fröhlich
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Fröhlich, S. (2012). Normal Coulomb Frames in \({\mathbb{R}}^{4}\) . In: Coulomb Frames in the Normal Bundle of Surfaces in Euclidean Spaces. Lecture Notes in Mathematics, vol 2053. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29846-2_3

Download citation

Publish with us

Policies and ethics

Search

Navigation