Constant mean curvature surfaces in homogeneously regular 3-manifolds.

*(English)*Zbl 1104.53057It is said that a Riemannian manifold \(M\) is homogeneously regular if there is a positive constant \(R\) such that any ball of radius \(R\) in \(M\) is embedded and all sectional curvatures inside the ball are bounded by some constant which is independent on the center of the ball. The author establishes some results concerning properly embedded constant mean curvature (cmc) surfaces in homogeneously regular three-manifolds under the assumption that the mean curvature is large. In particular, he proves that for a strongly stable cmc surface the distance from any its points to the boundary of the surface is bounded by some constant (for surfaces without boundary this implies their compactness), proves the maximum principle at infinity for properly embedded cmc surfaces, and establishes the bound for the genus of a closed (weakly) stable cmc surface (this bound equals three).

Reviewer: Iskander A. Taimanov (Novosibirsk)

##### MSC:

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |

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\textit{H. Rosenberg}, Bull. Aust. Math. Soc. 74, No. 2, 227--238 (2006; Zbl 1104.53057)

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##### References:

[1] | DOI: 10.1007/BF02564662 · Zbl 0679.53047 · doi:10.1007/BF02564662 |

[2] | DOI: 10.1007/BF01394782 · Zbl 0573.53038 · doi:10.1007/BF01394782 |

[3] | DOI: 10.1007/BF00147301 · Zbl 0724.53037 · doi:10.1007/BF00147301 |

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