Fuchsian Reduction (FR) is a general technique for reducing
nonlinear PDEs, near singularities, to an equation in which (i)
the singular set is characteristic and (ii) the equation is
asymptotically scale-invariant near the singular set. After an
overview of the main applications of FR, we outline the solution
of two problems:
\begin{enumerate}
\item The boundary regularity of the “conformal radius” on a
bounded domain in the plane, of class $C^{2+\alpha}$,
$0<\alpha<1$. This question was raised by Bandle and Flucher
(1996), motivated in particular by a result of Caffarelli and
Friedman (1985), for $C^{4+\alpha}$ strictly convex domains.
Similar results hold for the Loewner-Nirenberg problem. As a
consequence, we obtain a variational characterization of the
conformal radius, related to a new Hardy-type inequality.
\item A question raised by Fefferman and Graham (1985), on the
counterpart in even dimension of their construction of a
Ricci-flat metric with a homothety, associated to a Riemannian
manifold of odd dimension, generalizing the embedding of the unit
sphere in the null cone of Minkowski space. We also report on
recent results on the action of conformal changes on the metric.
\end{enumerate}