On stable constant mean curvature surfaces in $\mathbb S^2\times \mathbb R$ and $\mathbb H^2\times \mathbb R$
HTML articles powered by AMS MathViewer
- by Rabah Souam PDF
- Trans. Amer. Math. Soc. 362 (2010), 2845-2857 Request permission
Abstract:
We study the stability of immersed compact constant mean curvature (CMC) surfaces without boundary in some Riemannian 3-manifolds, in particular the Riemannian product spaces $\mathbb S^2 \times \mathbb R$ and $\mathbb H^2\times \mathbb R.$ We prove that rotational CMC spheres in $\mathbb H^2\times \mathbb R$ are all stable, whereas in $\mathbb S^2\times \mathbb R$ there exists some value $H_0\approx 0.18$ such that rotational CMC spheres are stable for $H\geq H_0$ and unstable for $0<H<H_0.$ We show that a compact stable immersed CMC surface in $\mathbb S^2\times \mathbb R$ is either a finite union of horizontal slices or a rotational sphere. In the more general case of an ambient manifold which is a simply connected conformally flat 3-manifold with nonnegative Ricci curvature we show that a closed stable immersed CMC surface is either a sphere or an embedded torus. Under the weaker assumption that the scalar curvature is nonnegative, we prove that a closed stable immersed CMC surface has genus at most three. In the case of $\mathbb H^2\times \mathbb R$ we show that a closed stable immersed CMC surface is a rotational sphere if it has mean curvature $H\ge 1/\sqrt {2}$ and that it has genus at most one if $1/\sqrt {3} < H < 1/\sqrt {2}$ and genus at most two if $H=1/\sqrt {3}.$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
- João Lucas Barbosa and Manfredo do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984), no. 3, 339–353. MR 731682, DOI 10.1007/BF01215045
- J. Lucas Barbosa, Manfredo do Carmo, and Jost Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z. 197 (1988), no. 1, 123–138. MR 917854, DOI 10.1007/BF01161634
- Shiu Yuen Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), no. 1, 43–55. MR 397805, DOI 10.1007/BF02568142
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- Joseph Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645–A1648 (French). MR 292357
- Wu-teh Hsiang and Wu-Yi Hsiang, On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces. I, Invent. Math. 98 (1989), no. 1, 39–58. MR 1010154, DOI 10.1007/BF01388843
- Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros, Proof of the double bubble conjecture, Ann. of Math. (2) 155 (2002), no. 2, 459–489. MR 1906593, DOI 10.2307/3062123
- Miyuki Koiso, Deformation and stability of surfaces with constant mean curvature, Tohoku Math. J. (2) 54 (2002), no. 1, 145–159. MR 1878932
- Peter Li and Shing Tung Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269–291. MR 674407, DOI 10.1007/BF01399507
- Barbara Nelli and Harold Rosenberg, Global properties of constant mean curvature surfaces in $\Bbb H^2\times \Bbb R$, Pacific J. Math. 226 (2006), no. 1, 137–152. MR 2247859, DOI 10.2140/pjm.2006.226.137
- Renato H. L. Pedrosa, The isoperimetric problem in spherical cylinders, Ann. Global Anal. Geom. 26 (2004), no. 4, 333–354. MR 2103404, DOI 10.1023/B:AGAG.0000047528.20962.e2
- Manuel Ritoré, Index one minimal surfaces in flat three space forms, Indiana Univ. Math. J. 46 (1997), no. 4, 1137–1153. MR 1631568, DOI 10.1512/iumj.1997.46.1299
- Antonio Ros, The isoperimetric problem, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 175–209. MR 2167260
- Antonio Ros, One-sided complete stable minimal surfaces, J. Differential Geom. 74 (2006), no. 1, 69–92. MR 2260928
- Antonio Ros, Stable periodic constant mean curvature surfaces and mesoscopic phase separation, Interfaces Free Bound. 9 (2007), no. 3, 355–365. MR 2341847, DOI 10.4171/IFB/168
- Antonio Ros and Rabah Souam, On stability of capillary surfaces in a ball, Pacific J. Math. 178 (1997), no. 2, 345–361. MR 1447419, DOI 10.2140/pjm.1997.178.345
- Antonio Ros and Enaldo Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56 (1995), no. 1, 19–33. MR 1338315, DOI 10.1007/BF01263611
- Marty Ross, Schwarz’ $P$ and $D$ surfaces are stable, Differential Geom. Appl. 2 (1992), no. 2, 179–195. MR 1245555, DOI 10.1016/0926-2245(92)90032-I
- Rabah Souam, On stability of stationary hypersurfaces for the partitioning problem for balls in space forms, Math. Z. 224 (1997), no. 2, 195–208. MR 1431192, DOI 10.1007/PL00004289
- R. Souam & E. Toubiana. On the classification and regularity of umbilic surfaces in homogeneous 3-manifolds. Mat. Contemp. 30 (2006), 201-215.
- Rabah Souam and Eric Toubiana, Totally umbilic surfaces in homogeneous 3-manifolds, Comment. Math. Helv. 84 (2009), no. 3, 673–704. MR 2507258, DOI 10.4171/CMH/177
- M. Spivak. A comprehensive introduction to differential geometry, Vol. 4. Publish or Perish, Boston, 1970.
- T. J. Willmore, Note on embedded surfaces, An. Şti. Univ. “Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.) 11B (1965), 493–496 (English, with Romanian and Russian summaries). MR 202066
Additional Information
- Rabah Souam
- Affiliation: Institut de Mathématiques de Jussieu, CNRS UMR 7586, Université Paris Diderot - Paris 7, Géométrie et Dynamique, Site Chevaleret, Case 7012, 75205 Paris Cedex 13, France
- Email: souam@math.jussieu.fr
- Received by editor(s): October 23, 2007
- Published electronically: January 20, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 2845-2857
- MSC (2010): Primary 53C42, 49Q10
- DOI: https://doi.org/10.1090/S0002-9947-10-04826-9
- MathSciNet review: 2592938