Local curvature estimates for the Ricci-harmonic flow

DOI10.1016/J.NA.2022.112961arXiv1810.09760MaRDI QIDQ6308640

Yi Li

Publication date: 23 October 2018

Abstract: In this paper we give an explicit bound of $Delt{a}_{g(t)}u(t)$ 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.

Mathematics Subject Classification ID

Differential geometric aspects of harmonic maps (53C43) Ricci flows (53E20)

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