Global properties of constant mean curvature surfaces in ℍ²× ℝ

Abstract

We discuss some aspects of the global behavior of surfaces in ‫ވ‬ 2 × ‫ޒ‬ with constant mean curvature H (known as H-surfaces). We prove a maximum principle at infinity for complete properly embedded H-surfaces with H > 1/ √ 2, and show that the genus of a compact stable H-surface with H > 1/ √ 2 is at most three.

On the First Stability Eigenvalue of Constant Mean Curvature Surfaces Into Homogeneous 3-Manifolds

“…As a consequence, we derive the following result which is related to Corollary 4.1 in [5] (it is important to realize that the notion of stability in Corollary 4.1 of [5], as well as in Theorem C of the same reference, is that of weakly stability although not explicitly stated). We study now the cases where the bundle curvature τ = 0.…”

Cylindrically bounded constant mean curvature surfaces in $\mathbb {H} ^2\times \mathbb {R}$

“…Thus, any properly embedded cmc surface which is cylindrically bounded has mean curvature H 0 > 1/2. This is a consequence of the half space theorem in H 2 × R (see [13] and [11], for example). So we can focus ourselves only on the H 0 > 1/2 case.…”

Finite Index Operators on Surfaces

“…To finish, we give some consequences of the above results for stable H−surface in E(κ, τ ). Nelli-Rosenberg [22] proved that there are no stable H−surfaces, H > 1/ √ 3, in H 2 × R, by proving a Distance Lemma. In general, thanks to a Distance Lemma, i.e.…”