We prove that a H -surface M in H 2 × R, |H | 1/2, inherits the symmetries of its boundary ∂ M, when ∂ M is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to π. Furthermore we prove that the asymptotic boundary of a surface with mean curvature bounded away from zero consists of parts of straight lines, provided it is sufficiently regular.
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.
Abstract. Let N n+1 be a Riemannian manifold with sectional curvatures uniformly bounded from below. When n = 3, 4, we prove that there are no complete (strongly) stable H-hypersurfaces, without boundary, provided |H| is large enough. In particular, we prove that there are no complete strongly stable H-hypersurfaces in R n+1 without boundary, H = 0.
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