发表机构
Politecnico di Milano; Università di Napoli Federico II（米兰理工大学; 那不勒斯费德里科二世大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明闭黎曼流形上临界 Ricci--Bourguignon 流（即 Schouten 流）的短时间存在性与唯一性，通过扩展系统解决退化性，并恢复原始流。
AI 中文摘要
设 $(M^n,g_0)$ 为闭光滑黎曼流形，其中 $n\geqslant 3$。我们证明了“临界”{\em Ricci--Bourguignon 流}的短时间存在性与唯一性：\begin{equation} \partial_t g=-2\operatorname{Ric}_g+\frac{R_g}{n-1}g\\,. \end{equation} 在时间常数重标度下，这等价于{\em Schouten 流}，因为方程右端为 $-2(n-2)$ 倍的 Schouten 张量。然而，在临界参数处，经过通常的“DeTurck 修正”后所得算子线性化的主符号仍具有零特征值。我们通过附加一个独立的标量未知量 $r$（用于表示数量曲率）来解决这一退化性，并考虑一个“扩展”系统，该系统由关于度量的严格抛物型方程与关于 $r$ 的带有曲率平方强迫项的输运-反应方程耦合而成。该扩展系统属于 Vol'pert 和 Khudyaev~\cite{VolpertKhudyaev1972} 研究的非线性复合抛物-双曲型系统类别。然后，对其定理进行标准的紧流形适配，即可得到扩展系统的唯一光滑解。最后，差值 $R_g-r$ 满足齐次线性抛物型方程，因此数量曲率约束“传播”，从而恢复原始临界 Ricci--Bourguignon 流。这解决了临界“Schouten”值 $\rho=1/(2(n-1))$ 处的短时间存在性与唯一性问题，该问题在~\cite{CatinoEtAl2017} 之前的 Ricci--Bourguignon 理论中未被解决。
英文摘要
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}.
CommentsThe strategy leading to the main argument was suggested during a series of interactions with ChatGPT 5.6 Pro. The authors subsequently checked, developed and independently verified all mathematical arguments and take full responsibility for the results and for the final manuscript
