Short-time existence for the Schouten flow
Let $(M^n,g_0)$ be a closed smooth Riemannian manifold, with $n\geqslant 3$. We prove short--time existence and uniqueness for the "critical" {\em Ricci--Bourguignon flow} \begin{equation} \partial_t g=-2\operatorname{Ric}_g+\frac{R_g}{n-1}g\,. \end{equation} Up to a constant rescaling of time, this is the {\em Schouten flow}, since the right--hand side of the equation is $-2(n-2)$ times the Schouten tensor. At the critical parameter, however, the principal symbol of the linearization of the operator that we obtain after the usual "DeTurck modification" still has a zero eigenvalue. We resolve this degeneracy by adjoining an independent scalar unknown $r$, intended to represent the scalar curvature, and considering an "extended" system consisting of a strictly parabolic equation for the metric coupled to a transport--reaction equation for $r$ with a curvature--square forcing. The extended system belongs to the class of nonlinear composite parabolic--hyperbolic systems studied by Vol'pert and Khudyaev~\cite{VolpertKhudyaev1972}. Then, a standard compact--manifold adaptation of their theorem yields a unique smooth solution of the extended system. Finally, the difference $R_g-r$ satisfies a homogeneous linear parabolic equation, so the scalar--curvature constraint "propagates" and the original critical Ricci--Bourguignon flow is recovered. This settles the short--time existence and uniqueness problem at the critical ``Schouten'' value $ρ=1/(2(n-1))$, which was left open by the previous Ricci--Bourguignon theory in~\cite{CatinoEtAl2017}.