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 $$K(P)\le -\frac{\varphi (\varphi -1)}{r(x{)}^2}$$ and a pointwise pinching condition $$|K(P)|\le C_K|K(P^{\prime })|$$ for some constants $\varphi >1$ and $C_K\ge 1$, where $P$ and $P^{\prime }$ are any 2-dimensional subspaces of $T_{x}M$ 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 $n>4/\varphi +1$.

To appear in J. Geom. Anal

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