Short-time existence for the Schouten flow
Abstract
Let (Mn,g0) be a closed smooth Riemannian manifold, with ngeqslant 3. We prove short--time existence and uniqueness for the "critical" {\em Ricci--Bourguignon flow} ∂t g=−2Ricg+Rg/(n−1)g . 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 Rg−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}.
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BibTeX
@misc{catino2026shorttime,
title = {{Short-time existence for the Schouten flow}},
author = {Giovanni Catino and Carlo Mantegazza},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06503},
eprint = {2609.06503},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06503},
}
RIS
TY - UNPB
TI - Short-time existence for the Schouten flow
AU - Giovanni Catino
AU - Carlo Mantegazza
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06503
UR - https://arxiv.org/abs/2609.06503
AB - Let (M^n,g_0) be a closed smooth Riemannian manifold, with ngeqslant 3. We prove short--time existence and uniqueness for the "critical" {\em Ricci--Bourguignon flow} ∂_t g=−2Ric_g+R_g/(n−1)g . 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}.
ER -