Second Ricci flow on noncompact Hermitian manifolds (Q2236636)

The current paper studies the second Ricci flows, a specialization of a type of flows first considered in [\textit{J. Streets} and \textit{G. Tian}, J. Eur. Math. Soc. (JEMS) 13, No. 3, 601--634 (2011; Zbl 1214.53055)]. In the following we suppose that \((M, g_0)\) is a complete noncompact Hermitian manifold and \(T\) denotes the torsion tensor of \(g_0\). The author obtains the following result of short-time existence (Theorem 1.1): \textbf{Theorem I.} If \(\sup_M|\mathrm{Rm}|+|T|^2+|\nabla T|<\infty, \) then the following flow has a short-time solution \(g=g(t)\) on \(M\times [0,\tau]\): \[ \frac{\partial}{\partial t}g = - S_g\ \text{ with }\ g(0)=g_0, \] where \(S_g = S_{i\bar j}\) is the second Ricci curvature with respect to the Chern connection of \(g\), i.e., \[ S_{i\bar j}= g^{k\bar l}\left(-\frac{\partial^2g_{k\bar l}}{\partial z^i\partial \bar z^j}+g^{r\bar s}\frac{\partial g_{k\bar s}}{\partial z^i}\frac{\partial g_{r\bar l}}{\partial \bar z^j}\right). \] Moreover, \(g(t)\) has bounded Chern curvature and torsion on \([0,\tau]\). The above result is then applied to obtain the following result about the existence of Kähler-Einstein metrics (part of Theorem 1.3): \textbf{Theorem II.} \(M\) admits a complete Kähler-Einstein metric \(g_{KE}=-\mathrm{Ric}(g_KE)\) with bounded curvature if for \((M,g_0)\) \begin{itemize} \item[(i)] the Riemannian curvature, the Chern curvature and the torsion are bounded, \item[(ii)] the Chern bisectional curvature is bounded, and \item[(iii)] the first Ricci curvature is quasinegative and is uniformly negative outside a compact set. \end{itemize} One aspect in which this result is different from various early results on the existence of complete Kähler-Einstein metrics is that \(M\) is not assumed to have a Kähler metric in advance. A brief overview of the proof of Theorem I is as follows. There is a constant \(\varepsilon_n>0\) (\(n=\dim\, M\)) validating the following: \begin{itemize} \item[(a)] (Section 5) Under certain assumption of bounded geometry on \((M,g_0)\) a solution \(g(t)\) exists on \([0, \tau(g_0,n)]\) for some \(\tau(g_0,n)>0\) and one has \[ (1+\varepsilon_n)^{-1}g_0 \leqslant g(t) \leqslant (1+\varepsilon_n)g_0.\tag{1} \] \item[(b)] (Section 4) If (1) holds on \(B_{g_0}(p,r+\delta)\times[0,\tau]\) for some \(p\in M\), \(r>0\), and \(\delta>0\), then one may obtain a priori estimates of \[ |\mathrm{Rm}|+|\nabla T|+|\nabla_{g_0}g(t)|\ \text{ and }\ |\mathrm{Rm}|+|\nabla T|+|\nabla_{g_0}g(t)|\ \text{ on }\ B_{g_0}(p,r+\delta)\times[0,\tau] \] and a priori estimates of \[ |\nabla^m\mathrm{Rm}|+|\nabla^m T|\ \text{ on }\ B_{g_0}(p,r+(m+1)^{-1}\delta)\times[0,\tau]\quad (m\in\mathbf N). \] \end{itemize} These estimates rely on expressions in Section 3 of \[ \partial_t g(t),\ (\partial_t-\Delta)\mathrm{tr}_{g(t)}g_0, \text{ and } (\partial_t-\Delta)|\nabla^{g(t)}-\nabla^{g_0}|^2_{g(t)}, \] as well as that of the time derivatives of the Chern-Ricci curvature and the higher derivatives of \(\mathrm{Rm}\) and \(T\). Following the idea of [\textit{M.-C. Lee} and \textit{L.-F. Tam}, J. Differ. Geom. 115, No. 3, 529--564 (2020; Zbl 1469.32017)], in Section 6, by summing up (a) and (b) yields a generalization of Theorem I by reduction to the case with bounded geometry. As for Theorem II, for a complete solution \(g(t)\) on \([0,\tau]\) with \(|\mathrm{Rm}|+|T|^2\) bounded on \(M\times [0,\tau]\), it is shown in Section 7 that one has \(\mathrm{Ric}(g(t))<0\) on \((0,\tau']\) for some \(\tau'>0\) if \(g_0\) has quasinegative Chern-Ricci curvature. In particular, \(M\) admits a possibly incomplete Kähler metric. This is used along with some estimates to show that, for some \(\tau>0\), \(-\mathrm{Ric}(g(\tau))\) is a complete Kähler metric uniformly equivalent to \(g(\tau)\), with certain estimates on \(|\nabla^m_{g(\tau)}h|\). This enables a usage of Theorem 5.1 of [\textit{J. Lott} and \textit{Z. Zhang}, Duke Math. J. 156, No. 1, 87--123 (2011; Zbl 1248.53050)] to conclude a complete Kähler-Einstein metric on \(M\).