Local curvature estimates for the Ricci-harmonic flow

TL;DR

This paper derives local curvature bounds for the Ricci-harmonic flow under Ricci curvature constraints, extends these estimates to generalized Ricci flows, and explores related curvature notions and uniqueness properties.

Contribution

It provides explicit local curvature estimates for Ricci-harmonic flow and extends these results to a class of generalized Ricci flows, connecting to recent conjectures and curvature notions.

Findings

Explicit bounds for $ riangle_{g(t)}u(t)$ and local curvature estimates.

Extension of curvature estimates to generalized Ricci flows.

Discussion of curvature tensor notions and flow uniqueness.

Abstract

In this paper we give an explicit bound of $\Delta_{g(t)}u(t)$ and the local curvature estimates for the Ricci-harmonic flow under the condition that the Ricci curvature is bounded along the flow. In the second part these local curvature estimates are extended to a class of generalized Ricci flow, introduced by the author \cite{LY1}, whose stable points give Ricci-flat metrics on a complete manifold, and which is very close to the $(K, N)$-super Ricci flow recently defined by Xiangdong Li and Songzi Li \cite{LL2014}. Next we propose a conjecture for Einstein's scalar field equations motivated by a result in the first part and the bounded $L^{2}$-curvature conjecture recently solved by Klainerman, Rodnianski and Szeftel \cite{KRS2015}. In the last two parts of this paper, we discuss two notions of "Riemann curvature tensor" in the sense of Wylie-Yeroshkin \cite{KW2017, KWY2017, Wylie2015, WY2016}, respectively, and Li \cite{LY3}, whose "Ricci curvature" both give the standard Bakey-\'Emery Ricci curvature \cite{BE1985}, and the forward and backward uniqueness for the Ricci-harmonic flow.