Let M be a complete embedded H-surface of bounded curvature in , a homogeneously regular 3-manifold. We prove that if H is large (in terms of the scalar curvature of N) then M is properly embedded. The proof follows from two theorems. First, if M is a complete stable immersed H-surface in N and H is large, then M is topologically a sphere. Secondly, a theorem of Meeks, Perez and Ros is used: limit leaves of CMC-laminations are stable.
Soit M une surface complète de courbure moyenne constante H dans , de courbure bornée. On montre que si H est grand (par rapport à la courbure scalaire de N) alors M est proprement plongée. La preuve utilise deux théorèmes. Le premier est que si M est est une surface de courbure moyenne constante stable dans N (complète) et si H est grand, alors M est topologiquement une sphère. Le second est un théorème de Meeks, Perez et Ros : Les feuilles limites d'une lamination CMC sont stables.
Accepted:
Published online:
Harold Rosenberg 1
@article{CRMATH_2009__347_3-4_183_0, author = {Harold Rosenberg}, title = {Remarks on surfaces of large mean curvature}, journal = {Comptes Rendus. Math\'ematique}, pages = {183--184}, publisher = {Elsevier}, volume = {347}, number = {3-4}, year = {2009}, doi = {10.1016/j.crma.2008.12.012}, language = {en}, }
Harold Rosenberg. Remarks on surfaces of large mean curvature. Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 183-184. doi : 10.1016/j.crma.2008.12.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.12.012/
[1] Limit leaves of a CMC lamination are stable (28 Jan 2008) | arXiv
[2] Constant mean curvature surfaces in homogeneously regular 3-manifolds, Bull. Austral. Math. Soc., Volume 74 (2006), pp. 227-238
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