Some criteria for parabolicity and hyperbolicity of the boundary sets of surfaces

We give criteria for the parabolicity and hyperbolicity of the boundary sets of surfaces $F=(D,ds^2_F)$, where $D$ is a domain in $\mathbb R^n$ and $ds^2_F$ is the square of the length element on $F$. We prove the parabolicity of certain boundary sets located on the graphs of the solutions of equations of minimal surface type. As an example we present a generalized maximum principle for the derivatives of solution of equations of minimal surface type where domains of $\mathbb R^n$ become “narrow” at infinity. We formulate criteria for the parabolicity and hyperbolicity of boundary sets on the graphs of spacelike surfaces in Minkowski space $\mathbb R_1^{n+1}$, and in particular, we obtain an essential strengthening of the theorem of Choi and Treibergs on the hyperbolicity of the graphs of entire solutions of the constant mean curvature equation in $\mathbb R_1^3$.