×

zbMATH — the first resource for mathematics

A Hopf differential for constant mean curvature surfaces in \(\mathbb S^2 \times \mathbb R\) and \(\mathbb H^2 \times\mathbb R\). (English) Zbl 1078.53053
A well-known theorem by H. Hopf states, that any immersed constant mean curvature (cmc) sphere \(S^2\to\mathbb{R}^3\) is in fact a standard distance sphere with radius \(1/H\), where \(H\) is this cmc. Here, the key is the discovery that the complexification of the traceless part of the second fundamental form \(h_\Sigma\) of an immersed surface \(\Sigma^2\) with constant \(H\) in \(\mathbb{R}^3\) is a holomorphic quadratic differential \(Q\) on \(\Sigma^2\). Afterwards this theorem has been extended replacing \(\mathbb{R}^3\) by \(\mathbb{S}^3\) or \(\mathbb{H}^3\). This differential \(Q\) played also a significant role in the discovery of the Wente tori and in the subsequent development of a general theory of cmc tori in \(\mathbb{R}^3\).
In the present paper a generalized quadratic differential \(Q\) is introduced for \(\Sigma^2\) in \(\mathbb{S}^2\times\mathbb{R}\) and \(\mathbb{H}^2\times\mathbb{R}\) (in other words, in \(M^2_\kappa\times\mathbb{R}\), where \(M^2_\kappa\) is of constant curvature \(\kappa\)). Hopf’s result is extended to cmc spheres in these target spaces. It is proved that such immersed spheres are surfaces of revolution. Four distinct classes of complete, possibly immersed, cmc surfaces \(\Sigma^2\to M^2_\kappa\times\mathbb{R}\) with vanishing quadratic differential \(Q\) are found. They are also geometrically described in detail. For \(\kappa< 0\) all four cases do actually occur, but if \(\kappa\geq 0\) then only the embedded cmc spheres \(S^2_H\subset M_\kappa\times\mathbb{R}\) occur.

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] Abresch, U., Constant mean curvature tori in terms of elliptic functions.J. Reine Angew. Math., 374 (1987), 169–192. · Zbl 0597.53003
[2] –, Old and new doubly periodic solutions of the sinh-Gordon equation, inSeminar on New Results in Nonlinear Partial Differential Equations (Bonn, 1984), pp. 37–73. Aspects Math., E10. Vieweg, Braunschweig, 1987.
[3] Alexandrov, A. D., Uniqueness theorems for surfaces in the large, V.Vestnik Leningrad. Univ., 13:19 (1958), 5–8 (Russian).
[4] –, Uniqueness theorems for surfaces in the large, I–V.Amer. Math. Soc. Transl., 21 (1962), 341–416.
[5] –, A characteristic property of spheres.Ann. Mat. Pura Appl., 58 (1962), 303–315. · Zbl 0107.15603
[6] Bobenko, A. I., All constant mean curvature tori inR 3,S 3,H 3 in terms of theta-functions.Math. Ann., 290 (1991), 209–245. · Zbl 0711.53007
[7] Burstall, F. E., Ferus, D., Pedit, F. &Pinkall, U., Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras.Ann. of Math., 138 (1993), 173–212. · Zbl 0796.53063
[8] Collin, P., Hauswirth, L. &Rosenberg, H., The geometry of finite topology Bryant surfaces.Ann. of Math., 153 (2001), 623–659. · Zbl 1066.53019
[9] Figueroa, C. B., Mercuri, F. &Pedrosa, R. H. L., Invariant surfaces of the Heisenberg groups.Ann. Mat. Pura Appl., 177 (1999), 173–194. · Zbl 0965.53042
[10] Gauss, C. F., Allgemeine Auflösung der Aufgabe: Die Theile einer gegebenen Fläche auf einer andern gegebenen Fläche so abzubilden, dass die Abbildung dem Abgebildeten in der Kleinsten Theilen, ähnlich wird (Kopenhagener Preisschrift).Astron. Abhandl., 3 (1825), 1–30;Phil. Mag., 4 (1828), 104–113, 206–215;Carl Friedrich Gauss Werke, Vierter Band, pp. 189–216. Der Königlichen Gesellschaft der Wissenschaften, Göttingen, 1873.
[11] Hauswirth, L., Roitman, P. &Rosenberg, H., The geometry of finite topology Bryant surfaces quasi-embedded in a hyperbolic manifold.J. Differential Geom., 60 (2002), 55–101. · Zbl 1067.53044
[12] Hitchin, N. J., Harmonic maps from a 2-torus to the 3-sphere.J. Differential Geom., 31 (1990), 627–710. · Zbl 0725.58010
[13] Hopf, H.,Differential Geometry in the Large. Lecture Notes in Math., 1000. Springer, Berlin, 1983.
[14] Hsiang, W.-T. &Hsiang, W.-Y., On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces, I.Invent. Math., 98 (1989), 39–58. · Zbl 0682.53057
[15] Hsiang, W.-Y., On soap bubbles and isoperimetric regions in noncompact symmetric spaces, I.Tôhoku Math. J., 44 (1992), 151–175. · Zbl 0819.53021
[16] Kapouleas, N., Complete embedded minimal surfaces of finite total curvature.J. Differential Geom., 47 (1997), 95–169. · Zbl 0936.53006
[17] de Lira, J., To appear.
[18] Mazzeo, R. &Pacard, F., Constant mean curvature surfaces with Delaunay ends.Comm. Anal. Geom., 9 (2001), 169–237. · Zbl 1005.53006
[19] Nelli, B. &Rosenberg, H., Minimal surfaces inH 2 {\(\times\)}R.Bull. Braz. Math. Soc., 33 (2002), 263–292. · Zbl 1038.53011
[20] Pedrosa, R. H. L. &Ritoré, M., 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. · Zbl 0956.53049
[21] Pinkall, U. &Sterling, I., On the classification of constant mean curvature tori.Ann. of Math., 130 (1989), 407–451. · Zbl 0683.53053
[22] Rosenberg, H., Bryant surfaces, inThe Global Theory of Minimal Surfaces in Flat Spaces (Martina Franca, 1999), pp. 67–111. Lecture Notes in Math., 1775. Springer, Berlin, 2002.
[23] –, Minimal surfaces inM 2 {\(\times\)}R.Illinois J. Math., 46 (2002), 1177–1195. · Zbl 1036.53008
[24] Wells, R. O.,Differential Analysis on Complex Manifolds, 2nd edition. Graduate Texts in Math., 65. Springer, New York-Berlin, 1980. · Zbl 0435.32004
[25] Wente, H., Counterexample to a conjecture of H. Hopf.Pacific J. Math., 121 (1986), 193–243. · Zbl 0586.53003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.