Variational Methods in Geometry

Survey Article
  • 437 Downloads

Abstract

Variational principles are ubiquitous in nature. Many geometric objects such as geodesics or minimal surfaces allow variational characterizations. We recall some basic ideas in the calculus of variations, also relevant for some of the most advanced research in the field today, and show how a subtle variation of standard methods can lead to surprising improvements, with numerous applications.

Mathematics Subject Classification

49-02 53-02 58-02 58E05 58E10 58E12 49Q05 58E20 

References

  1. 1.
    Agol, I., Marques, F.C., Neves, A.: Min-max theory and the energy of links. J. Am. Math. Soc. 29(2), 561–578 (2016) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ballmann, W.: Der Satz von Lusternik und Schnirelmann. In: Beiträge zur Differentialgeometrie, Heft 1. Bonner Math. Schriften, vol. 102, pp. 1–25. Univ. Bonn, Bonn (1978) Google Scholar
  3. 3.
    Ballmann, W., Ziller, W.: On the number of closed geodesics on a compact Riemannian manifold. Duke Math. J. 49(3), 629–632 (1982) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bangert, V.: Geodätische Linien auf Riemannschen Mannigfaltigkeiten. Jahresber. Dtsch. Math.-Ver. 87(2), 39–66 (1985) MathSciNetMATHGoogle Scholar
  5. 5.
    Bangert, V.: On the existence of closed geodesics on two-spheres. Int. J. Math. 4(1), 1–10 (1993) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bethuel, F., Brezis, H., Hélein, F.: Ginzburg-Landau Vortices. Progress in Nonlinear Differential Equations and their Applications, vol. 13. Birkhäuser, Boston (1994) CrossRefMATHGoogle Scholar
  7. 7.
    Birkhoff, G.D.: Dynamical systems with two degrees of freedom. Trans. Am. Math. Soc. 18(2), 199–300 (1917) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Böhme, R., Tromba, A.J.: The index theorem for classical minimal surfaces. Ann. Math. (2) 113(3), 447–499 (1981) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Borer, F., Galimberti, L., Struwe, M.: “Large” conformal metrics of prescribed Gauss curvature on surfaces of higher genus. Comment. Math. Helv. 90(2), 407–428 (2015) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Brezis, H., Coron, J.-M.: Multiple solutions of H-systems and Rellich’s conjecture. Commun. Pure Appl. Math. 37(2), 149–187 (1984) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Buttazzo, G., Giaquinta, M., Hildebrandt, S.: One-Dimensional Variational Problems. Oxford Lecture Series in Mathematics and its Applications, vol. 15. Clarendon/Oxford Univ. Press, New York (1998) MATHGoogle Scholar
  12. 12.
    Carlotto, A., Malchiodi, A.: Weighted barycentric sets and singular Liouville equations on compact surfaces. J. Funct. Anal. 262(2), 409–450 (2012) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Colin de Verdière, Y.: Spectrum of the Laplace operator and periodic geodesics: thirty years after. Ann. Inst. Fourier 57(7), 2429–2463 (2007) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Coron, J.-M.: Topologie et cas limite des injections de Sobolev. C. R. Math. Acad. Sci. Paris, Sér. I 299(7), 209–212 (1984) MathSciNetMATHGoogle Scholar
  15. 15.
    Courant, R.: Dirichlet’s Principle, Conformal Mappings, and Minimal Surfaces. Interscience, New York (1950) MATHGoogle Scholar
  16. 16.
    del Pino, M., Felmer, P.L.: On the basic concentration estimate for the Ginzburg-Landau equation. Differ. Integral Equ. 11(5), 771–779 (1998) MathSciNetMATHGoogle Scholar
  17. 17.
    del Pino, M., Román, C.: Large conformal metrics with prescribed sign-changing Gauss curvature. Calc. Var. Partial Differ. Equ. 54(1), 763–789 (2015) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O.: Minimal Surfaces. Part I: Boundary Value Problems. Grundlehren der mathematischen Wissenschaften, vol. 295. Springer, Berlin (1992) MATHGoogle Scholar
  19. 19.
    Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O.: Minimal Surfaces. Part II: Boundary Regularity. Grundlehren der mathematischen Wissenschaften, vol. 296. Springer, Berlin (1992) MATHGoogle Scholar
  20. 20.
    Ding, W.Y., Liu, J.Q.: A note on the problem of prescribing Gaussian curvature on surfaces. Trans. Am. Math. Soc. 347(3), 1059–1066 (1995) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Djadli, Z., Malchiodi, A.: Existence of conformal metrics with constant Q-curvature. Ann. Math. (2) 168(3), 813–858 (2008) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Douglas, J.: Solution of the problem of Plateau. Trans. Am. Math. Soc. 33(1), 263–321 (1931) MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Douglas, J.: Minimal surfaces of higher topological structure. Ann. Math. (2) 40(1), 205–298 (1939) MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Galimberti, L.: Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation on the torus. Calc. Var. Partial Differ. Equ. 54(3), 2483–2501 (2015) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Goldstine, H.H.: A History of the Calculus of Variations from the 17th Through the 19th Century. Studies in the History of Mathematics and Physical Sciences, vol. 5. Springer, New York (1980) MATHGoogle Scholar
  26. 26.
    Hildebrandt, S.: Boundary behavior of minimal surfaces. Arch. Ration. Mech. Anal. 35, 47–82 (1969) MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Hildebrandt, S.: Calculus of variations today, reflected in the Oberwolfach meetings. In: Perspectives in Mathematics, pp. 321–336. Birkhäuser, Basel (1984) Google Scholar
  28. 28.
    Hildebrandt, S., Tromba, A.: The Parsimonious Universe. Shape and Form in the Natural World. Copernicus, New York (1996) CrossRefMATHGoogle Scholar
  29. 29.
    Hildebrandt, S., von der Mosel, H.: On two-dimensional parametric variational problems. Calc. Var. Partial Differ. Equ. 9(3), 249–267 (1999) MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Huber, H.: Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. Math. Ann. 138, 1–26 (1959) MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Imbusch, C., Struwe, M.: Variational principles for minimal surfaces. In: Topics in Nonlinear Analysis. Progress in Nonlinear Differential Equations and their Applications, vol. 35, pp. 477–498. Birkhäuser, Basel (1999) CrossRefGoogle Scholar
  32. 32.
    Jost, J., Struwe, M.: Morse-Conley theory for minimal surfaces of varying topological type. Invent. Math. 102(3), 465–499 (1990) MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Kazdan, J.L., Warner, F.W.: Curvature functions for compact 2-manifolds. Ann. Math. (2) 99, 14–47 (1974) MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Klingenberg, W.: Lectures on Closed Geodesics. Grundlehren der mathematischen Wissenschaften, vol. 230. Springer, Berlin (1978) MATHGoogle Scholar
  35. 35.
    Lamm, T.: Energy identity for approximations of harmonic maps from surfaces. Trans. Am. Math. Soc. 362(8), 4077–4097 (2010) MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Lusternik, L., Schnirelman, L.: Sur le problème de trois géodésiques fermées sur les surfaces de genre 0. C. R. Math. Acad. Sci. Paris 189, 269–271 (1929) MATHGoogle Scholar
  37. 37.
    Lyusternik, L., Schnirelman, L.: Topological methods in variational problems and their application to the differential geometry of surfaces. Usp. Mat. Nauk 2(1), 166–217 (1947) (in Russian) MathSciNetGoogle Scholar
  38. 38.
    Marques, F.C., Neves, A.: Rigidity of min-max minimal spheres in three-manifolds. Duke Math. J. 161(14), 2725–2752 (2012) MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Marques, F.C., Neves, A.: Min-max theory and the Willmore conjecture. Ann. Math. (2) 179(2), 683–782 (2014) MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Michelat, A., Rivière, T.: A viscosity method for the min-max construction of closed geodesics. arXiv:1511.04545
  41. 41.
    Morse, M.: The Calculus of Variations in the Large. American Mathematical Society Colloquium Publications, vol. 18. Am. Math. Soc., New York (1934). Reprint: Am. Math. Soc. Providence, 1996 MATHGoogle Scholar
  42. 42.
    Morse, M., Tompkins, C.: The existence of minimal surfaces of general critical types. Ann. Math. (2) 40(2), 443–472 (1939) MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Nitsche, J.C.C.: The boundary behavior of minimal surfaces. Kellogg’s theorem and branch points on the boundary. Invent. Math. 8, 313–333 (1969) MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Nitsche, J.C.C.: Vorlesungen über Minimalflächen. Grundlehren der mathematischen Wissenschaften, vol. 199. Springer, Berlin (1975) CrossRefMATHGoogle Scholar
  45. 45.
    Palais, R.S.: Morse theory on Hilbert manifolds. Topology 2, 299–340 (1963) MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Palais, R.S., Smale, S.: A generalized Morse theory. Bull. Am. Math. Soc. 70, 165–172 (1964) MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Radó, T.: On Plateau’s problem. Ann. Math. (2) 31(3), 457–469 (1930) MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Rivière, T.: A viscosity method for the min-max theory of minimal surfaces. arXiv:1508.07141
  49. 49.
    Rupflin, M.: Teichmüller harmonic map flow from cylinders. Math. Ann. (2016). doi:10.1007/s00208-016-1456-4. arXiv:1501.07552 Google Scholar
  50. 50.
    Rupflin, M., Topping, P.M.: Flowing maps to minimal surfaces. Am. J. Math. 138(4), 1095–1115 (2016) MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. (2) 113(1), 1–24 (1981) MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Struwe, M.: On a critical point theory for minimal surfaces spanning a wire in \(\mathbb{R}^{n}\). J. Reine Angew. Math. 349, 1–23 (1984) MathSciNetMATHGoogle Scholar
  53. 53.
    Struwe, M.: Nonuniqueness in the Plateau problem for surfaces of constant mean curvature. Arch. Ration. Mech. Anal. 93(2), 135–157 (1986) MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Struwe, M.: Plateau’s Problem and the Calculus of Variations. Mathematical Notes, vol. 35. Princeton Univ. Press, Princeton (1988) MATHGoogle Scholar
  55. 55.
    Struwe, M.: The existence of surfaces of constant mean curvature with free boundaries. Acta Math. 160(1–2), 19–64 (1988) MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Struwe, M.: Existence of periodic solutions of Hamiltonian systems on almost every energy surface. Bol. Soc. Bras. Mat. (N. S.) 20(2), 49–58 (1990) MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Struwe, M.: Multiple solutions to the Dirichlet problem for the equation of prescribed mean curvature. In: Analysis, et Cetera, pp. 639–666. Academic Press, Boston (1990) CrossRefGoogle Scholar
  58. 58.
    Struwe, M.: Une estimation asymptotique pour le modèle de Ginzburg-Landau. C. R. Math. Acad. Sci. Paris, Sér. I 317(7), 677–680 (1993) MathSciNetGoogle Scholar
  59. 59.
    Struwe, M.: On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2 dimensions. Differ. Integral Equ. 7(5–6), 1613–1624 (1994). Erratum: Differ. Integral Equ. 8(1), 224 (1995) MathSciNetMATHGoogle Scholar
  60. 60.
    Struwe, M.: Positive solutions of critical semilinear elliptic equations on non-contractible planar domains. J. Eur. Math. Soc. 2(4), 329–388 (2000) MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Struwe, M.: Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34. Springer, Berlin (2008) MATHGoogle Scholar
  62. 62.
    Tromba, A.J.: Degree theory on oriented infinite-dimensional varieties and the Morse number of minimal surfaces spanning a curve \(\mathbb{R}^{n}\). Part II: \(n=3\). Manuscr. Math. 48(1–3), 139–161 (1984) CrossRefMATHGoogle Scholar
  63. 63.
    Tromba, A.J.: Degree theory on oriented infinite-dimensional varieties and the Morse number of minimal surfaces spanning a curve in \(\mathbb{R}^{n}\). Part I: \(n\ge 4\). Trans. Am. Math. Soc. 290(1), 385–413 (1985) MathSciNetMATHGoogle Scholar
  64. 64.
    Wente, H.C.: Large solutions to the volume constrained Plateau problem. Arch. Ration. Mech. Anal. 75(1), 59–77 (1980) MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Zhou, X.: Min-max hypersurface in manifold of positive Ricci curvature. arXiv:1504.00966

Copyright information

© Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.MathematikETH-ZürichZürichSwitzerland

Personalised recommendations