Existence and non-existence for a mean curvature equation in hyperbolic space
Elias M. Guio, Ricardo Sá Earp · Communications on Pure & Applied Analysis · 2005
There exists a well-known criterion for the solvability of theDirichlet Problem for the constant mean curvature equation in bounded smooth domains in Euclidean space. This classicalresult was established by Serrin in 1969. Focusing the DirichletProblem for radial vertical graphs P.-A. Nitsche has establishedan existence and non-existence results on account of a criterionbased on the notion of a hyperbolic cylinder. In this work wecarry out a similar but distinct result in hyperbolic spaceconsidering a different Dirichlet Problem based on another systemof coordinates. We consider a non standard cylindergenerated by horocycles cutting orthogonally a geodesic plane$\mathcal P$ along the boundary of a domain $\Omega\subset \mathcal P.$ We provethat a non strict inequality between the mean curvature $\mathcal H'_{\mathcal C}(y)$of this cylinder along $\partial \Omega$ and the prescribed meancurvature $\mathcal H(y),$ i.e $\mathcal H'_{\mathcal C}(y)\geq |\mathcal H(y)|, \forally\in\partial\Omega$ yields existence of our Dirichlet Problem. Thuswe obtain existence of surfaces whose graphs have prescribed meancurvature $\mathcal H(x)$ in hyperbolic space taking a smooth prescribedboundary data $\varphi.$ This result is sharp because if ourcondition fails at a point $y$ a non-existence result can beinferred.