Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Willmore two-spheres in the four-sphere

Author: Sebastián Montiel
Journal: Trans. Amer. Math. Soc. 352 (2000), 4469-4486
MSC (2000): Primary 53C40; Secondary 53A10, 53C28
Published electronically: June 13, 2000
MathSciNet review: 1695032
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Genus zero Willmore surfaces immersed in the three-sphere $\mathbb{S}^3$correspond via the stereographic projection to minimal surfaces in Euclidean three-space with finite total curvature and embedded planar ends. The critical values of the Willmore functional are $4\pi k$, where $k\in\mathbb{N}^*$, with $k\ne 2,3,5,7$. When the ambient space is the four-sphere $\mathbb{S}^4$, the regular homotopy class of immersions of the two-sphere $\mathbb{S}^2$ is determined by the self-intersection number $q\in\mathbb{Z}$; here we shall prove that the possible critical values are $4\pi (\vert q\vert+k+1)$, where $k\in\mathbb{N}$. Moreover, if $k=0$, the corresponding immersion, or its antipodal, is obtained, via the twistor Penrose fibration $\mathbb{P}^3\rightarrow \mathbb{S}^4$, from a rational curve in $\mathbb{P}^3$and, if $k\ne 0$, via stereographic projection, from a minimal surface in $\mathbb{R}^4$ with finite total curvature and embedded planar ends. An immersion lies in both families when the rational curve is contained in some $\mathbb{P}^2\subset\mathbb{P}^3$ or (equivalently) when the minimal surface of $\mathbb{R}^4$ is complex with respect to a suitable complex structure of $\mathbb{R}^4$.

References [Enhancements On Off] (What's this?)

  • [Bl] W. Blaschke, Vorlesungen Uber Differentialgeometrie III, Springer, Berlin, 1929.
  • [Br1] R.L. Bryant, A duality theorem for Willmore surfaces, J. of Diff. Geom., 20(1984), 23-53. MR 86j:58029
  • [Br2] R.L. Bryant, Surfaces in conformal geometry, Proc. of Symp. in Pure Math., 48(1988), 227-240. MR 89m:53102
  • [CU] I. Castro, F. Urbano, Lagrangian surfaces in the complex Euclidean plane with conformal Maslov form, Tôhoku Math. J., 45(1993), 565-582. MR 94j:53064
  • [E] N. Ejiri, Willmore surfaces with a duality in $S^N(1)$, Proc. London Math. Soc., 57(1988), 383-416. MR 89h:53117
  • [ES] J. Eells, S. Salamon, Twistorial constructions of harmonic maps of surfaces into four-manifolds, Ann. Scuola Norm. Sup. Pisa, 12(1985), 589-640. MR 87i:58042
  • [F] T. Friedrich, On surfaces in four-spaces, Ann. Global Anal. Geom., 2(1984), 257-287. MR 86h:53061
  • [GH] P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978. MR 80b:14001
  • [Hi] M.W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc., 93(1959), 242-276. MR 22:9980
  • [HO] D. Hoffman, R. Osserman, The geometry of the generalized Gauss map, Mem. Amer. Math. Soc., 236(1980). MR 82b:53012
  • [KM] J.L. Koszul, B. Malgrange, Sur certaines structures fibres complexes, Arch. Math., 9(1958), 102-109. MR 24:A1729
  • [K1] R. Kusner, Conformal geometry and complete minimal surfaces, Bull. Amer. Math. Soc., 17(1987), 291-295. MR 88j:53008
  • [K2] R. Kusner, Comparison surfaces for the Willmore problem, Pacific Math. J., 138(1989), 317-345. MR 90e:53013
  • [KS] R. Kusner, N. Schmitt, The spinor representation of surfaces in space, preprint.
  • [LS] R. Lashof, S. Smale, On immersions of manifolds in Euclidean space, Ann. of Math., 68(1958), 562-583. MR 21:2246
  • [LY] P. Li, S.T. Yau, A new conformal invariant and its applications to the Willmore conjecture and first eigenvalue of compact surfaces, Invent. Math., 69(1982), 269-291. MR 84f:53049
  • [MR] S. Montiel, A. Romero, Complex Einstein hypersurfaces of indefinite complex space forms, Math. Proc. Cambridge Phil. Soc., 93(1983), 495-508. MR 85f:53049
  • [MRo] S. Montiel, A. Ros, Minimal immersions of surfaces by the first eigenfunctions and conformal area, Invent. Math., 83(1986), 153-166. MR 87d:53109
  • [R] A. Ros, The Willmore conjecture in the real projective space, Math. Res. Lett. 6 (1999), 487-493.
  • [RT] H. Rosenberg, E. Toubiana, Some remarks on deformations of minimal surfaces, Trans. Amer. Math. Soc., 295(1986), 491-499. MR 88a:53005b
  • [S] S. Salamon, Topics in four-dimensional Riemannian geometry, Lecture Notes in Mathematics, 1022(1982), 34-124. MR 85i:53002
  • [We] J.L. Weiner, On an inequality of P. Wintgen for the integral of the square of the mean curvature, J. London Math. Soc., 34(1986), 148-158. MR 87k:53147
  • [Wei] A. Weinstein, Lectures on symplectic manifolds, Conf. Board Math. Sci., Regional Conf. Ser. Math., vol. 29, Amer. Math. Soc., Providence, RI, 1977. MR 57:4244
  • [Wi1] T.J. Willmore, Note on embedded surfaces, An. Sti. Univ. "Al. I. Cuza" Iasi Sect. I a Mat., 11(1965), 493-496. MR 34:1940
  • [Wi2] T.J. Willmore, Mean curvature of Riemannian immersions, J. London Math. Soc., 3(1971), 307-310. MR 44:959
  • [Win] P. Wintgen, On the total curvature of surfaces in $\mathbb{E}^4$, Colloq. Math., 39(1978), 289-296. MR 80f:53037
  • [Wo] J.A. Wolf, Spaces of Constant Curvature, McGraw-Hill, New York, 1967. MR 36:829

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C40, 53A10, 53C28

Retrieve articles in all journals with MSC (2000): 53C40, 53A10, 53C28

Additional Information

Sebastián Montiel
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, Spain

Keywords: Willmore surface, minimal surface
Received by editor(s): September 30, 1998
Received by editor(s) in revised form: March 15, 1999
Published electronically: June 13, 2000
Additional Notes: Research partially supported by a DGICYT grant PB97-0785
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society