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

TL;DR

This paper investigates the solvability of the minimal graph equation on certain negatively curved manifolds with specific curvature bounds and pinching conditions, solving the Dirichlet problem for high enough dimensions.

Contribution

It establishes solvability of the asymptotic Dirichlet problem for the minimal graph equation under pointwise pinching and curvature bounds on Cartan-Hadamard manifolds, extending previous results.

Findings

Solves the Dirichlet problem for dimensions n > 4/φ + 1.

Provides conditions on curvature bounds for solvability.

Extends known results to manifolds with specific pinching conditions.

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 $n>4/\phi+1$.