Properly embedded surfaces with prescribed mean curvature in $${\mathbb {H}}^2\times {\mathbb {R}}$$
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Properly embedded surfaces with prescribed mean curvature in ℍ2 × ℝ Antonio Bueno1 Received: 20 April 2020 / Accepted: 25 September 2020 © Springer Nature B.V. 2020
Abstract The aim of this paper is to extend classic results of the theory of constant mean curvature surfaces in the product space ℍ2 × ℝ to the class of immersed surfaces whose mean curvature is given as a C1 function depending on their angle function. We cover topics such as the existence of a priori curvature and height estimates for graphs and a structure-type result, which classifies properly embedded surfaces with finite topology and at most one end. Keywords Prescribed mean curvature · Product space · Height estimate · Structure result Mathematics Subject Classification 53A10
1 Introduction A fundamental result due to Meeks [14] states that there do not exist properly embedded surfaces in ℝ3 with positive constant mean curvature (CMC in the following) having finite topology and one end. The proof relies in the existence of uniform height estimates for CMC graphs and the possibility of applying Alexandrov reflection technique with respect to any direction of ℝ3. The ideas introduced by Meeks have been adapted to further three dimensional spaces and curvature functions rather than the mean curvature. For instance, Korevaar, Kusner, Meeks and Solomon [12] exhibited this result for surfaces with CMC H0 > 1 in the hyperbolic space ℍ3 . Regarding other homogeneous spaces, √ Meeks’ theorem was generalized to the space ℍ2 × ℝ for surfaces with CMC H0 > 1∕ 3 by Nelli-Rosenberg [16], and for all values H0 > 1∕2 by Espinar-Gálvez-Rosenberg (see Corollary 6.2 and Theorem 7.2 in [8]). A similar result in ℍ2 × ℝ was obtained in [8] for surfaces with positive, constant extrinsic curvature: The author was partially supported by MICINN-FEDER Grant No. MTM2016-80313-P and Junta de Andalucía Grant No. FQM325. * Antonio Bueno [email protected] 1
Departamento de Geometría y Topología, Universidad de Granada, E‑18071 Granada, Spain
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Vol.:(0123456789)
Annals of Global Analysis and Geometry
Theorem 1.1 ([8, 16]) Let be H0 > 1∕2 . There do not exist properly embedded surfaces with CMC equal to H0 in ℍ2 × ℝ having finite topology and one end. The condition H0 > 1∕2 is known as the critical value for the mean curvature in ℍ2 × ℝ and has a strong geometric implication. If 0 < H0 ≤ 1∕2 there exist entire CMC vertical graphs, or equivalently, there exist CMC spheres if and only if H0 > 1∕2. Regarding more general curvature functions, Meeks’ theorem has been recently generalized in ℝ3 for the following class of surfaces: given H ∈ C1 (𝕊2 ) , a surface Σ in ℝ3 is an H-surface if its mean curvature HΣ is given at every p ∈ Σ by HΣ (p) = H(Np ) , where N ∶ Σ → 𝕊2 is the Gauss map of Σ . The definition of this class of surfaces has its motivation in the famous Minkowski problem for ovaloids, and the existence and uniqueness of H-spheres were studied, among others, by Alexandrov and Pogorelov in the ’50s [2]. In [4, 5], the author jointly with Gálvez and Mira has star
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