Osaka Journal of Mathematics

Willmore-like functionals for surfaces in 3-dimensional Thurston geometries

Dmitry Berdinsky and Yuri Vyatkin

Full-text: Open access


We find analogues of the Willmore functional for each of the Thurston geometries with $4$--dimensional isometry group such that the CMC--spheres in these geometries are critical points of these functionals.

Article information

Osaka J. Math., Volume 54, Number 1 (2017), 75-83.

First available in Project Euclid: 3 March 2017

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 57R70: Critical points and critical submanifolds


Berdinsky, Dmitry; Vyatkin, Yuri. Willmore-like functionals for surfaces in 3-dimensional Thurston geometries. Osaka J. Math. 54 (2017), no. 1, 75--83.

Export citation


  • U. Abresch and H. Rosenberg: A Hopf differential for constant mean curvature surfaces in $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$, Acta Math. 193 (2004), 141–174.
  • U. Abresch and H. Rosenberg: Generalized Hopf differentials, Mat. Contemp. 28 (2005), 1–28.
  • M. Belkhelfa, F. Dillen and J. Inoguchi: Surfaces with parallel second fundamental form in Bianchi–Cartan–Vranceanu spaces, PDE's, submanifolds and affine differential geometry, Banach center publ., Polish Acad. Sci. 57 (2002), 67–87.
  • D. Berdinsky: On some generalization of the Willmore functional for surfaces in $\widetilde{S\!L}_2$, Siberian Electronic Mathematical Reports 7 (2010), 140–145.
  • D. Berdinsky and I. Taimanov: Surfaces in three–dimensional Lie groups, Siberian Math. Journal 46 (2005), 1005–1019.
  • D. Berdinsky and I. Taimanov: Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional, Siberian Math. Journal 48 (2007), 395–407.
  • R. Caddeo, P. Piu and A. Ratto: SO(2)–invariant minimal and constant mean curvature surfaces in $3$–dimensional homogeneous spaces, Manuscripta Math. 87 (1995), 1–12.
  • B. Daniel: Isometric immersions into 3–dimensional homogeneous manifolds, Comment. Math. Helv. 82 (2007), 87–131.
  • J.F. Dorfmeister, J. Inoguchi and S. Kobayashi: A loop group method for minimal surfaces in the three–dimensional Heisenberg group, Asian Journal of Mathematics (to appear), arXiv:1210.7300v4 [math.DG].
  • I. Fernández and P. Mira: Constant mean curvature surfaces in 3–dimensional Thurston geometries, Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), 830–861. Hindustan Book Agency (HBA), India, 2012.
  • C. Figueroa, F. Mercuri and R. Pedrosa: Invariant surfaces of the Heisenberg groups, Ann. Math. Pura Appl. 177 (1999), 173–194.
  • W.Y. Hsiang and W.T. Hsiang: On the uniqueness of isoperimetric solutions and embedded soap bubbles in noncompact symmetric spaces, Invent. Math. 98 (1989), 39–58.
  • G. Huisken and A. Polden: Geometric evolution equations for hypersurfaces, In Calculus of variations and geometric evolution problems: lectures given at the 2nd session of the Centro Internazionale Estivo (C.I.M.E.) held in Cetraro, Italy, June 15-22, 1996, 45–84. Springer, 1999.
  • R.H.L. Pedrosa and M. Ritoré: Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems, Indiana Univ. Math. J. 48 (1999), 1357–1394.
  • C. Peñafiel: Invariant surfaces in $\widetilde{PS\!L}{}_2(\mathbb{R}, \tau)$ and applications, Bull. Braz. Math. Soc., New Series 43 (2012), 545–578.
  • P. Scott: The geometries of 3–manifolds, Bull. London Math. Soc. 15 (1983), 401–487.
  • P. Tomter: Constant mean curvature surfaces in the Heisenberg group, Proceedings of Symposia in Pure Mathematics 54 (1993), 485–495.
  • F. Torralbo: Rotationally invariant constant mean curvature surfaces in homogeneous 3–manifolds, Diff. Geom. Appl. 28 (2010), 593–607.
  • J.L. Weiner: On a problem of Chen, Willmore, et al, Indiana Univ. Math. J. 27, (1978), 19–35.