Solvability of minimal graph equation under pointwise pinching condition for sectional curvatures

Abstract

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan-Hadamard manifold $M$ whose radial sectional curvatures outside a compact set satisfy an upper bound $$K(P)\le - \frac{\phi(\phi-1)}{r(x)^2}$$ and a pointwise pinching condition $$|K(P)|\le C_K|K(P')|$$ for some constants $\phi>1$ and $C_K\ge 1$, where $P$ and $P'$ are any 2-dimensional subspaces of $T_xM$ containing the (radial) vector $\nabla r(x)$ and $r(x)=d(o,x)$ is the distance to a fixed point $o\in M$. We solve the asymptotic Dirichlet problem with any continuous boundary data for dimensions $n>4/\phi+1$.