The Michigan Mathematical Journal

New Examples of Constant Mean Curvature Surfaces in S2×R and H2×R

José M. Manzano and Francisco Torralbo

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Article information

Michigan Math. J., Volume 63, Issue 4 (2014), 701-723.

First available in Project Euclid: 5 December 2014

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Mathematical Reviews number (MathSciNet)

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]


Manzano, José M.; Torralbo, Francisco. New Examples of Constant Mean Curvature Surfaces in $\mathbb{S}^{2}\times\mathbb{R}$ and $\mathbb{H}^{2}\times\mathbb{R}$. Michigan Math. J. 63 (2014), no. 4, 701--723. doi:10.1307/mmj/1417799222.

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  • [AEG08] J. A. Aledo, J. M. Espinar, and J. A. Gálvez, Height estimates for surfaces with positive constant mean curvature in $\mathbb{M}^{2}\times\mathbb{R}$, Illinois J. Math. 52 (2008), no. 1, 203–211.
  • [Aron57] N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235–249.
  • [CS85] H. Choi and R. Schoen, The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature, Invent. Math. 81 (1985), no. 3, 387–394.
  • [CH13] M. Chuaqui and L. Hauswirth, Generalized Krust theorem in homogeneous spaces, preprint.
  • [Dan07] B. Daniel, Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv. 82 (2007), 87–131.
  • [DHM09] B. Daniel, L. Hauswirth, and P. Mira, Lectures notes on homogeneous 3-manifolds, 4th KIAS workshop on differential geometry, Korea Institute for Advanced Study, Seoul, Korea, 2009.
  • [DHKW92] U. Dierkes, S. Hildebrandt, A. Küster, and O. Wohlrab, Minimal surfaces, vols. I and II, Springer-Verlag, Berlin, 1992.
  • [GB93] K. Große-Brauckmann, New surfaces of constant mean curvature, Math. Z. 214 (1993), no. 4, 527–565.
  • [GB05] K. Große-Brauckmann, Cousins of constant mean curvature surfaces, Global theory of minimal surfaces, Clay Math. Proc., 2, pp. 747–767, Amer. Math. Soc., Providence, RI, 2005.
  • [HST08] L. Hauswirth, R. Sa Earp, and E. Toubiana, Associate and conjugate minimal immersions in $\mathbf{M}\times\mathbf{R}$, Tohoku Math. J. (2) 60 (2008), no. 2, 267–286.
  • [HH89] W. Hsiang and W. Hsiang, On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces I, Invent. Math. 98 (1989), no. 1, 39–58.
  • [K89] H. Karcher, The triply periodic minimal surfaces of A. Schoen and their constant mean curvature companions, Manuscripta Math. 64 (1989), 291–357.
  • [KPS88] H. Karcher, U. Pinkall, and I. Sterling, New minimal surfaces in ${S^{3}}$, J. Differential Geom. 28 (1988), no. 2, 169–185.
  • [Law70] H. B. Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374.
  • [Man12] J. M. Manzano, Superficies de curvatura media constante en espacios homogéneos, Ph.D. thesis, Universidad de Granada, 2012. ISBN 978-84-90282694.
  • [Man13] H. B. Lawson Jr., Estimates for constant mean curvature graphs in $M\times\mathbb{R}$, Rev. Mat. Iberoam. 29 (2013), 1263–1281.
  • [MRR] L. Mazet, M. M. Rodríguez, and H. Rosenberg, Periodic constant mean curvature surfaces in $\mathbb{H}^{2}\times\mathbb{R}$, Asian J. Math. 18 (2014), no. 5, 829–858.
  • [MY82] W. Meeks and S.-T. Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179 (1982), 151–168.
  • [MR12] F. Morabito and M. M. Rodríguez, Saddle towers and minimal $k$-noids in $\mathbb{H}^{2}\times\mathbb{R}$, J. Inst. Math. Jussieu 11 (2012), no. 2, 333–349.
  • [PR99] R. 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), no. 4, 1357–1394.
  • [P] A. L. Pinheiro, Minimal vertical graphs in Heisenberg space, preprint.
  • [Po94] K. Polthier, Geometric a priori estimates for hyperbolic minimal surfaces, Bonner Math. Schriften 263 (1994).
  • [R] M. M. Rodríguez, Minimal surfaces with limit ends in $\mathbb{H}^{2}\times \mathbb{R}$, J. Reine Angew. Math. 685 (2013), 123–141.
  • [ST09] R. Souam and E. Toubiana, Totally umbilic surfaces in homogeneous 3-manifolds, Comment. Math. Helv. 84 (2009), no. 3, 673–704.
  • [Thu] W. Thurston, Three-dimensional geometry and topology, Princeton Math. Ser., 35, Princeton University Press, Princeton, 1997.
  • [Tor10a] F. Torralbo, Compact minimal surfaces in the Berger spheres, Ann. Global Anal. Geom. 41 (2012), 391–405.
  • [Tor10b] F. Torralbo, Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds, Differential Geom. Appl. 28 (2010), no. 5, 593–607.