  The planar embedding conjecture asserts that any planar metric admits an
embedding into L_1 with constant distortion. This is a well-known open problem
with important algorithmic implications, and has received a lot of attention
over the past two decades. Despite significant efforts, it has been verified
only for some very restricted cases, while the general problem remains elusive.
  In this paper we make progress towards resolving this conjecture. We show
that every planar metric of non-positive curvature admits a constant-distortion
embedding into L_1. This confirms the planar embedding conjecture for the case
of non-positively curved metrics.
