Annals of Global Analysis and Geometry

, Volume 56, Issue 1, pp 147–165 | Cite as

On the variation of curvature functionals in a space form with application to a generalized Willmore energy

  • Anthony GruberEmail author
  • Magdalena Toda
  • Hung Tran


Functionals involving surface curvature are important across a range of scientific disciplines, and their extrema are representative of physically meaningful objects such as atomic lattices and biomembranes. Inspired in particular by the relationship of the Willmore energy to lipid bilayers, we consider a general functional depending on a surface and a symmetric combination of its principal curvatures, and provided the surface is immersed in a 3-D space form of constant sectional curvature. We calculate the first and second variations of this functional, extending known results and providing computationally accessible expressions given entirely in terms of the basic geometric information found in the surface fundamental forms. Further, we motivate and introduce the p-Willmore energy functional, applying the stability criteria afforded by our calculations to prove a result about the p-Willmore energy of spheres.


Curvature functionals Variational problems Surface immersions Willmore energy 

Mathematics Subject Classification

58E12 53C42 53A05 53A35 



  1. 1.
    Aleksandrov, V.A.: On the integral mean curvature of nonrigid surfaces. Sibirsk. Mat. Zh. 50(5), 963–966 (2009). MathSciNetGoogle Scholar
  2. 2.
    Almgren Jr, F.J., Rivin, I.: The mean curvature integral is invariant under bending. Geom. Topol. Monogr. 1, 1–21 (1998). MathSciNetzbMATHGoogle Scholar
  3. 3.
    Atanasov, V., Dandoloff, R.: Quantum-elastic bump on a surface. Euro. J. Phys. 38(1), 015405 (2016). zbMATHGoogle Scholar
  4. 4.
    Athukorallage, B., Bornia, G., Paragoda, T., Toda, M.: Willmore-type energies and Willmore-type surfaces in space forms. JP J. Geom.Topol. 18(2), 93 (2015)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Barbosa, J.L., do Carmo, M.P., Eschenburg, J.: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Z. 197(1), 123–138 (1988). MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bryant, R.L.: A duality theorem for Willmore surfaces. J. Differ. Geom. 20(1), 23–53 (1984)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Canham, P.B.: The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theo. Biol. 26(1), 61–81 (1970). Google Scholar
  8. 8.
    Capovilla, R.: Elastic bending energy: a variational approach. J. Geom. Symmetry Phys. 45, 1–45 (2017). MathSciNetzbMATHGoogle Scholar
  9. 9.
    do Carmo, M.P.: Riemannian geometry. Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston (1992). zbMATHGoogle Scholar
  10. 10.
    Clelland, J.N.: From Frenet to Cartan: the method of moving frames. Graduate studies in mathematics. American Mathematical Society (2017). Accessed 11 Aug 2018
  11. 11.
    Dalphin, J., Henrot, A., Masnou, S., Takahashi, T.: On the minimization of total mean curvature. J Geom Anal 26(4), 2729–2750 (2016). MathSciNetzbMATHGoogle Scholar
  12. 12.
    Doğan, G., Nochetto, R.H.: First variation of the general curvature-dependent surface energy. ESAIM Math. Model. Numer. Anal. 46(1), 59–79 (2011). MathSciNetzbMATHGoogle Scholar
  13. 13.
    Durand, L.: Stability and oscillations of a soap film: an analytic treatment. Am. J. Phys. 49(4), 334–343 (1981). MathSciNetGoogle Scholar
  14. 14.
    Dziuk, G.: Computational parametric Willmore flow. Numer. Math. 111(1), 55 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Elliott, C.M., Fritz, H., Hobbs, G.: Small deformations of Helfrich energy minimising surfaces with applications to biomembranes. Math. Models Methods Appl. Sci. 27(08), 1547–1586 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Filgueiras, C., Silva, E.O., Andrade, F.M.: Nonrelativistic quantum dynamics on a cone with and without a constraining potential. J. Math. Phys. 53(12), 122106 (2012). MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gálvez, J.A.: Surfaces of constant curvature in 3-dimensional space forms. Mat. Contemp. 37, 1–42 (2009)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Germain, S.: Recherches sur la théorie des surfaces élastiques. Mme. Ve. Courcier (1821). Accessed 3 Oct 2018
  19. 19.
    Guo, Z.: Generalized Willmore functionals and related variational problems. Differ. Geom. Appl. 25(5), 543–551 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Helfrich, W.: Elastic properties of lipid bilayers: Theory and possible experiments. Zeitschrift für Naturforschung C 28(11–12), 693–703 (1973). Google Scholar
  21. 21.
    Joshi, P., Séquin, C.: Energy minimizers for curvature-based surface functionals. Comput. Aided Des. Appl. 4(5), 607–617 (2007). Google Scholar
  22. 22.
    Kik, R.A.: Molecular modeling of proteinlike inclusions in lipid bilayers: lipid-mediated interactions. Phys. Rev. E (2010). Google Scholar
  23. 23.
    Lautrup, B.: Physics of continuous matter: exotic and everyday phenomena in the macroscopic world. CRC Press, Boca Raton (2011)zbMATHGoogle Scholar
  24. 24.
    Mantoulidis, C., Miao, P.: Total mean curvature, scalar curvature, and a variational analog of Brown-York mass. Commun. Math. Phys. 352(2), 703–718 (2017). MathSciNetzbMATHGoogle Scholar
  25. 25.
    Marques, F.C., Neves, A.: Min-max theory and the Willmore conjecture. Ann. Math. 179(2), 683–782 (2014)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Marques, F.C., Neves, A.: The Willmore conjecture. Jahresbericht der Deutschen Mathematiker-Vereinigung 116(4), 201–222 (2014). MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mondino, A., Nguyen, H.T.: A gap theorem for Willmore tori and an application to the Willmore flow. Nonlinear Anal. Theory Methods Appl. 102, 220–225 (2014). MathSciNetzbMATHGoogle Scholar
  28. 28.
    Paragoda, T.: Application of the moving frame method to deformed Willmore surfaces in space forms. J. Geom. Phys. 128, 199–208 (2018). MathSciNetzbMATHGoogle Scholar
  29. 29.
    Séquin, C.H., Chang, P.Y., Moreton, H.P.: Scale-Invariant Functional for smooth curves and surfaces. In: Hagen H., Farin G., Noltemeier H. (eds) Geometric modelling. computing supplement, vol 10. Springer, Vienna (1995).
  30. 30.
    Shubin, M.A.: Pseudodifferential operators and spectral theory. Springer, Berlin (2001). zbMATHGoogle Scholar
  31. 31.
    Siegel, D.P.: Determining the ratio of the Gaussian curvature and bending elastic moduli of phospholipids from QII phase unit cell dimensions. Biophys. J. 91(2), 608–618 (2006). Google Scholar
  32. 32.
    Sjöstrand, F.S., Andersson-Cedergren, E., Dewey, M.M.: The ultrastructure of the intercalated discs of frog, mouse and guinea pig cardiac muscle. J. Ultrastruct. Res. 1(3), 271–287 (1958). Google Scholar
  33. 33.
    Toda, M., Athukorallage, B.: Geometric models for secondary structures in proteins. In: Mladenov IM, Ludu A, Yoshioka A (eds) Geometry, integrability and quantization XVI, pp. 282–300. Avangard Prima, Sofia (2015).
  34. 34.
    Toda, M., Zhang, F., Athukorallage, B.: Elastic surface model for beta-barrels: geometric, computational, and statistical analysis. Proteins Struct. Funct. Bioinform. 86(1), 35–42 (2017). Google Scholar
  35. 35.
    Tu, L.: Differential Geometry: Connections, Curvature, and Characteristic Classes. Graduate Texts in Mathematics. Springer International Publishing (2017). Accessed 13 Aug 2018
  36. 36.
    Tu, Z.C., Ou-Yang, Z.C.: A geometric theory on the elasticity of bio-membranes. J. Phys. A Math. Gen. 37(47), 11407–11429 (2004). MathSciNetzbMATHGoogle Scholar
  37. 37.
    Viswanathan, K., Parthasarathy, R.: A conformal field theory of extrinsic geometry of 2-d surfaces. Ann. Phys. 244(2), 241–261 (1995). MathSciNetzbMATHGoogle Scholar
  38. 38.
    Weiner, J.L.: On a problem of Chen, Willmore, et al. Indiana Univ. Math. J. 27(1), 19–35 (1978)MathSciNetzbMATHGoogle Scholar
  39. 39.
    White, J.H.: A global invariant of conformal mappings in space. Proc. Am. Math. Soc. 38(1), 162 (1973). MathSciNetzbMATHGoogle Scholar
  40. 40.
    Willmore, T.J.: Note on embedded surfaces. An. Sti. Univ.“Al. I. Cuza” Iasi Sect. I a Mat.(NS) B 11, 493–496 (1965)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Yang, Y.: Topological bounds on bending energy for lipid vesicles. Phys. Rev. E (2018). Google Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsTexas Tech UniversityLubbockUSA

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