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Existence and non-existence for a mean curvature equation in hyperbolic space

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  • There exists a well-known criterion for the solvability of the Dirichlet Problem for the constant mean curvature equation in bounded smooth domains in Euclidean space. This classical result was established by Serrin in 1969. Focusing the Dirichlet Problem for radial vertical graphs P.-A. Nitsche has established an existence and non-existence results on account of a criterion based on the notion of a hyperbolic cylinder. In this work we carry out a similar but distinct result in hyperbolic space considering a different Dirichlet Problem based on another system of coordinates. We consider a non standard cylinder generated by horocycles cutting orthogonally a geodesic plane $\mathcal P$ along the boundary of a domain $\Omega\subset \mathcal P.$ We prove that a non strict inequality between the mean curvature $\mathcal H'_{\mathcal C}(y)$ of this cylinder along $\partial \Omega$ and the prescribed mean curvature $\mathcal H(y),$ i.e $\mathcal H'_{\mathcal C}(y)\geq |\mathcal H(y)|, \forall y\in\partial\Omega$ yields existence of our Dirichlet Problem. Thus we obtain existence of surfaces whose graphs have prescribed mean curvature $\mathcal H(x)$ in hyperbolic space taking a smooth prescribed boundary data $\varphi.$ This result is sharp because if our condition fails at a point $y$ a non-existence result can be inferred.
    Mathematics Subject Classification: 35J25, 53A10.

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