An extension of Ruh-Vilms’ theorem to hypersurfaces in symmetric spaces and some applications
HTML articles powered by AMS MathViewer
- by Álvaro Ramos and Jaime Ripoll PDF
- Trans. Amer. Math. Soc. 368 (2016), 4731-4749 Request permission
Abstract:
This paper has two main purposes: First, to extend a well-known theorem of Ruh-Vilms in the Euclidean space to symmetric spaces and, secondly, to apply this result to extend the Hoffman-Osserman-Schoen theorem (HOS theorem) to $3$-dimensional symmetric spaces. Precisely, we define a Gauss map of a hypersurface $M^{n-1}$ immersed in a symmetric space $N^{n}$ taking values in the unit pseudo-sphere $\mathbb {S}^m$ of the Lie algebra $\mathfrak {g}$ of the isometry group of $N$, $\dim {\mathfrak {g}}=m+1,$ and it is proved that $M$ has CMC if and only if its Gauss map is harmonic. As an application, it is proved that if $\dim {N}=3$ and the image of the Gauss map of a CMC surface $S$ immersed in $N$ is contained in a hemisphere of $\mathbb {S}^m$ determined by a vector $X$, then $S$ is invariant by the one-parameter subgroup of isometries of $N$ of the Killing field determined by $X$. In particular, an extension of the HOS theorem to the $3$-dimensional hyperbolic space is obtained, which, as far as the authors know, has not been done.
It is also shown that the holomorphic quadratic form induced by the Gauss map coincides (up to a sign) with the Hopf quadratic form when the ambient space is $\mathbb {H}^{3}, \mathbb {R}^{3}$ and $\mathbb {S}^{3}$ and coincides with the Abresch-Rosenberg quadratic form when the ambient space is $\mathbb {H}^{2}\times \mathbb {R}$ and $\mathbb {S}^{2}\times \mathbb {R}$. This then provides a unified way of relating Hopf’s and Abresch-Rosenberg’s quadratic form with the quadratic form induced by a harmonic Gauss map of a CMC surface in these five spaces.
References
- Uwe Abresch and Harold Rosenberg, A Hopf differential for constant mean curvature surfaces in $\textbf {S}^2\times \textbf {R}$ and $\textbf {H}^2\times \textbf {R}$, Acta Math. 193 (2004), no. 2, 141–174. MR 2134864, DOI 10.1007/BF02392562
- Uwe Abresch and Harold Rosenberg, Generalized Hopf differentials, Mat. Contemp. 28 (2005), 1–28. MR 2195187
- Henrique Araújo and Maria Luiza Leite, Surfaces in $\Bbb S^2\times \Bbb R$ and $\Bbb H^2\times \Bbb R$ with holomorphic Abresch-Rosenberg differential, Differential Geom. Appl. 29 (2011), no. 2, 271–278. MR 2784305, DOI 10.1016/j.difgeo.2010.12.010
- Fidelis Bittencourt and Jaime Ripoll, Gauss map harmonicity and mean curvature of a hypersurface in a homogeneous manifold, Pacific J. Math. 224 (2006), no. 1, 45–63. MR 2231651, DOI 10.2140/pjm.2006.224.45
- Shiing Shen Chern, On surfaces of constant mean curvature in a three-dimensional space of constant curvature, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 104–108. MR 730266, DOI 10.1007/BFb0061413
- Benoît Daniel, Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv. 82 (2007), no. 1, 87–131. MR 2296059, DOI 10.4171/CMH/86
- Benoît Daniel, The Gauss map of minimal surfaces in the Heisenberg group, Int. Math. Res. Not. IMRN 3 (2011), 674–695. MR 2764875, DOI 10.1093/imrn/rnq092
- B. Daniel, I. Fernandez, and P. Mira, The Gauss map of surfaces in $\widetilde {PSL}_{2}(\mathbb {R})$, preprint, arXiv:1305.1491.
- Benoît Daniel and Laurent Hauswirth, Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group, Proc. Lond. Math. Soc. (3) 98 (2009), no. 2, 445–470. MR 2481955, DOI 10.1112/plms/pdn038
- Benoît Daniel and Pablo Mira, Existence and uniqueness of constant mean curvature spheres in $\textrm {Sol}_3$, J. Reine Angew. Math. 685 (2013), 1–32. MR 3181562, DOI 10.1515/crelle-2012-0016
- J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, DOI 10.1112/blms/10.1.1
- N. do Espírito-Santo, S. Fornari, K. Frensel, and J. Ripoll, Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric, Manuscripta Math. 111 (2003), no. 4, 459–470. MR 2002821, DOI 10.1007/s00229-003-0357-5
- José M. Espinar and Harold Rosenberg, Complete constant mean curvature surfaces and Bernstein type theorems in $M^2\times \Bbb R$, J. Differential Geom. 82 (2009), no. 3, 611–628. MR 2534989
- Isabel Fernández and Pablo Mira, Harmonic maps and constant mean curvature surfaces in $\Bbb H^2\times \Bbb R$, Amer. J. Math. 129 (2007), no. 4, 1145–1181. MR 2343386, DOI 10.1353/ajm.2007.0023
- Isabel Fernández and Pablo Mira, A characterization of constant mean curvature surfaces in homogeneous 3-manifolds, Differential Geom. Appl. 25 (2007), no. 3, 281–289. MR 2330457, DOI 10.1016/j.difgeo.2006.11.006
- Isabel Fernández and Pablo Mira, Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space, Trans. Amer. Math. Soc. 361 (2009), no. 11, 5737–5752. MR 2529912, DOI 10.1090/S0002-9947-09-04645-5
- Doris Fischer-Colbrie and Richard Schoen, The structure of complete stable minimal surfaces in $3$-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199–211. MR 562550, DOI 10.1002/cpa.3160330206
- Susana Fornari and Jaime Ripoll, Killing fields, mean curvature, translation maps, Illinois J. Math. 48 (2004), no. 4, 1385–1403. MR 2114163
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original. MR 1834454, DOI 10.1090/gsm/034
- D. A. Hoffman, R. Osserman, and R. Schoen, On the Gauss map of complete surfaces of constant mean curvature in $\textbf {R}^{3}$ and $\textbf {R}^{4}$, Comment. Math. Helv. 57 (1982), no. 4, 519–531. MR 694604, DOI 10.1007/BF02565874
- Heinz Hopf, Differential geometry in the large, Lecture Notes in Mathematics, vol. 1000, Springer-Verlag, Berlin, 1983. Notes taken by Peter Lax and John Gray; With a preface by S. S. Chern. MR 707850, DOI 10.1007/978-3-662-21563-0
- Maria Luiza Leite and Jaime Ripoll, On quadratic differentials and twisted normal maps of surfaces in $\Bbb S^2\times \Bbb R$ and $\Bbb H^2\times \Bbb R$, Results Math. 60 (2011), no. 1-4, 351–360. MR 2836904, DOI 10.1007/s00025-011-0151-8
- L. A. Masal′tsev, A version of the Ruh-Vilms theorem for surfaces of constant mean curvature in $S^3$, Mat. Zametki 73 (2003), no. 1, 92–105 (Russian, with Russian summary); English transl., Math. Notes 73 (2003), no. 1-2, 85–96. MR 1993542, DOI 10.1023/A:1022126101717
- William H. Meeks III, Constant mean curvature spheres in $\rm Sol_3$, Amer. J. Math. 135 (2013), no. 3, 763–775. MR 3068401, DOI 10.1353/ajm.2013.0025
- W. H. Meeks III, P. Mira, J. Pérez, and A. Ros, Constant mean curvature spheres in homogeneous three-spheres, preprint.
- W. H. Meeks III, P. Mira, J. Pérez, and A. Ros, Constant mean curvature spheres in homogeneous three-manifolds, Work in progress.
- William H. Meeks III and Joaquín Pérez, Constant mean curvature surfaces in metric Lie groups, Geometric analysis: partial differential equations and surfaces, Contemp. Math., vol. 570, Amer. Math. Soc., Providence, RI, 2012, pp. 25–110. MR 2963596, DOI 10.1090/conm/570/11304
- Ernst A. Ruh and Jaak Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569–573. MR 259768, DOI 10.1090/S0002-9947-1970-0259768-5
Additional Information
- Álvaro Ramos
- Affiliation: Departamento de Matematica, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, 91501-970, Porto Alegre, Rio Grande do Sul, Brazil
- Email: alvaro.ramos@ufrgs.br
- Jaime Ripoll
- Affiliation: Departamento de Matematica, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, 91501-970, Porto Alegre, Rio Grande do Sul, Brazil
- MR Author ID: 148575
- Email: jaime.ripoll@ufrgs.br
- Received by editor(s): May 16, 2014
- Published electronically: September 15, 2015
- Additional Notes: This research was supported by CNPq - Brasil
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 4731-4749
- MSC (2010): Primary 53C42
- DOI: https://doi.org/10.1090/tran6667
- MathSciNet review: 3456159