Browsing by Subject "Mean curvature equation"

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  • Casteras, Jean-Babtiste; Heinonen, Esko; Holopainen, Ilkka; Lira, Jorge (2020)
    We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature H in warped product manifolds Mx rho R. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on H and the mean curvature of the Killing cylinders over geodesic spheres. In the process we obtain a uniform interior gradient estimate improving previous results by Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to 0 or to -infinity provided that H satisfies certain bounds with respect to the sectional curvatures of M and the norm of the Killing vector field. Finally we obtain non-existence results if the prescribed mean curvature function H grows too fast.
  • Casteras, Jean-Baptiste; Heinonen, Esko; Holopainen, Ilkka (2020)
    We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold M with only one end if M has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and p-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures.