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 \(n>4/\phi+1\).