We prove a version of the Jenkins-Serrin theorem for the existence of CMC graphs over bounded domains with infinite boundary data in Sol 3 . Moreover, we construct examples of admissible domains where the results may be applied.

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In a Hadamard manifold M , it is proved that if u is a λ-eigenfunction of the Laplacian that belongs to L p (M ) for some p ≥ 2, then u is bounded and u ∞ ≤ C u p , where C depends only on p, λ and on the dimension of M . This result is obtained in the more general context of a complete Riemannian manifold endowed with an isoperimetric function H satisfying some integrability condition. In this case, the constant C depends on p, λ and H.

Let Ω ⊂ R 2 an unbounded convex domain and H > 0 be given, there exists a graph G ⊂ R 3 of constant mean curvature H over Ω with ∂G = ∂Ω if and only if Ω is included in a strip of width 1/H [7,12]. In this paper we obtain results in H 2 × R in the same direction: given H ∈ (0, 1/2), if Ω is included in a region of H 2 ×{0} bounded by two equidistant hypercycles ℓ(H) apart, we show that, if the geodesic curvature of ∂Ω is bounded from below by −1, then there is an H-graph G over Ω with ∂G = ∂Ω. We also present more refined existence results involving the curvature of ∂Ω, which can also be less than −1.

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