We show, by modifying Borbelys example, that there are $3$-dimen-sional Cartan-Hadamard manifolds $M$, with sectional curvatures $le -1$, such that the asymptotic Dirichlet problem for a class of quasilinear elliptic PDEs, including the minimal graph equation, is not solvable.
It is given a topological pinching for the injectivity radius of a compact embedded surface either in the sphere or in the hyperbolic space
A classical problem in constant mean curvature hypersurface theory is, for given $Hgeq 0$, to determine whether a compact submanifold $Gamma^{n-1}$ of codimension two in Euclidean space $R_+^{n+1}$, having a single valued orthogonal projection on $R^
n$, is the boundary of a graph with constant mean curvature $H$ over a domain in $R^n$. A well known result of Serrin gives a sufficient condition, namely, $Gamma$ is contained in a right cylinder $C$ orthogonal to $R^n$ with inner mean curvature $H_Cgeq H$. In this paper, we prove existence and uniqueness if the orthogonal projection $L^{n-1}$ of $Gamma$ on $R^n$ has mean curvature $H_Lgeq-H$ and $Gamma$ is contained in a cone $K$ with basis in $R^n$ enclosing a domain in $R^n$ containing $L$ such that the mean curvature of $K$ satisfies $H_Kgeq H$. Our condition reduces to Serrins when the vertex of the cone is infinite.
