The global geometry of surfaces with prescribed mean curvature in ℝ³

Abstract

We develop a global theory for complete hypersurfaces in R n+1 whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces in R n+1 , and also that of self-translating solitons of the mean curvature flow. For the particular case n = 2, we will obtain results regarding a priori height and curvature estimates, non-existence of complete stable surfaces, and classification of properly embedded surfaces with at most one end. Basic properties of H-hypersurfaces

Invariant hypersurfaces with linear prescribed mean curvature

“…Let us consider an oriented hypersurface Σ immersed into R n+1 whose mean curvature is denoted by H Σ and its Gauss map by η : Σ → S n ⊂ R n+1 . Following [BGM1], given a function H ∈ C 1 (S n ), Σ is said to be a hypersurface of prescribed mean curvature H if…”

“…Nevertheless, the global geometry of complete, non-compact hypersurfaces of prescribed mean curvature in R n+1 has been unexplored for general choices of H until recently. In this framework, the first author jointly with Gálvez and Mira have started to develop the global theory of hypersurfaces with prescribed mean curvature in [BGM1], taking as a starting point the well-studied global theory of CMC hypersurfaces in R n+1 . The same authors have also studied rotational hypersurfaces in R n+1 , getting a Delaunaytype classification result and several examples of rotational hypersurfaces with further symmetries and topological properties (see [BGM2]).…”

“…These isometries and the symmetries inherited by the invariant H λ -hypersurfaces are induced to Equation (1.2) easing the treatment of its solutions. We must emphasize that, although the authors already defined the class of immersed H λ -hypersurfaces in [BGM1], the classification of neither cylindrical nor rotational H λ -hypersurfaces in [BGM2] was covered.…”

Rotational hypersurfaces of prescribed mean curvature

“…So, as a direct application of Theorems 4.1 and 4.2, and since there are no compact H-hypersurfaces in R n+1 if H ∈ C 1 (S n ) vanishes somewhere (see [BGM,Proposition 2.6]), we have: Corollary 4.3 Let H ∈ C 1 (S 2 ) be even (i.e. H(x) = H(−x) ∀x ∈ S 2 ) and rotationally symmetric, and let Σ be an immersed H-surface in R 3 diffeomorphic to S 2 .…”

Half-space theorems for properly immersed surfaces in $${\mathbb {R}}^{3}$$ with prescribed mean curvature

“…Besides the milestones reached concerning the uniqueness of ovaloids with prescribed mean curvature, the global properties of immersed surfaces in R 3 governed by Eq. (1.1) remained largely unexplored until the author in joint work with Gálvez and Mira [2,3] developed the global theory of surfaces with prescribed mean curvature. In [3], we studied the structure of properly embedded H-surfaces and curvature estimates for stable H-surfaces in R 3 .…”