Abstract
Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy \({\mathcal{W}} = \int H^2\) under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the critical points of arbitrary geometric functionals on the space of immersions under the constraint that the admissible variations infinitesimally preserve the conformal structure. Besides constrained Willmore surfaces we discuss in some detail examples of constrained minimal and volume critical surfaces, the critical points of the area and enclosed volume functional under the conformal constraint.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blaschke, W.: Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln. Grundlehren 29. Springer, Berlin (1929)
Bohle, C., Peters, G.P.: Soliton Spheres (2007) (in preparation)
Bohle, C., Peters, G.P.: Bryant surfaces with smooth ends, arXiv:math.DG/0411480
Bohle, C.: Constrained Willmore tori in the 4-sphere (2007) (in preparation)
Bryant, R.L.: A duality theorem for Willmore surfaces. J. Diff. Geom. 20, 23–53 (1984)
Bryant, R.L.: Surfaces in conformal geometry. The mathematical heritage of Hermann Weyl (Durham, NC, 1987). In: Proceedings of symposium in pure mathematics, Vol. 48, pp. 227–240. American Mathematical Society, Providence (1988)
Burstall, F., Pedit, F., Pinkall, U.: Schwarzian derivatives and flows of surfaces. Contemp. Math. 308, 39–61 (2002) arXiv: math.DG/0111169
Early, C.J., Eells, J.: A fibre bundle description of Teichmüller theory. J. Diff. Geom. 3, 19–43 (1969)
Germain, S.: Recherches sur la théorie des surfaces élastiques. Courcier, Paris (1821)
Garsia, A.M.: An imbedding of closed Riemann surfaces in Euclidean space. Comment. Math. Helv. 35, 93–110 (1961)
Langer, J., Singer, D.A.: The total squared curvature of closed curves. J. Differ. Geom. 20, 1–22 (1984)
Pinkall, U.: Hopf tori in S 3. Invent. Math. 81, 379–386 (1985)
Pinkall, U., Sterling, I.: Willmore surfaces. Math. Intelligencer 9, 38–43 (1987)
Richter, J.: Conformal maps of a Riemann surface into the space of quaternions. Thesis, TU-Berlin (1997)
Rüedy, R.A.: Embeddings of open Riemann surfaces. Comment. Math. Helv. 46, 214–225 (1971)
Schmidt, M.: A proof of the Willmore conjecture, arXiv: math.DG/0203224
Sziegoleit, F.: Bedingt minimale Flächen. Diplomarbeit, TU-Berlin (2004)
Tromba, A.J.: Teichmüller Theory in Riemannian Geometry. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1992)
Weiner J.L. (1978) On a problem of Chen, Willmore et al. Indiana Univ. Math. J. 27, 19–35
Willmore, T.J.: Note on embedded surfaces. An. Şti. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 11, 493–496 (1965)
Willmore, T.J.: Riemannian Geometry. Oxford University Press, Oxford (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
C. Bohle, G. P. Peters and U. Pinkall are partially supported by DFG SPP 1154.
Rights and permissions
About this article
Cite this article
Bohle, C., Peters, G.P. & Pinkall, U. Constrained Willmore surfaces. Calc. Var. 32, 263–277 (2008). https://doi.org/10.1007/s00526-007-0142-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-007-0142-5