Some existence results on the exterior Dirichlet problem for the minimal hypersurface equation (Q540343)

The authors study the existence of solutions for the minimal hypersurface equation in an unbounded domain of a complete noncompact Riemannian manifold \(M\) and subject to a Dirichlet boundary condition. The existence of solutions in the cases that either \(M\) has nonnegative Ricci curvature or if \(M\) is simply connected and with sectional curvature \(K_M\) satisfying \(K_M\leq -k^2 < 0\) is established for some positive constant \(k\). The proofs combine the maximum principle, the Hessian comparison theorem and mean curvature estimates of some related level hypersurfaces.