\(C^\alpha\)-compactness and the Calabi flow on Kähler surfaces with negative scalar curvature (Q5954526)

Let \(M\) be a compact Kähler \(m\)-manifold that has a Kähler metric \(ds^2= g_{\alpha\overline\beta} dz^\alpha\otimes dz^\beta\); let \(\Omega_0= [\omega_0]\) be a fixed Kähler class on \(M\) for \(\omega_0= {\sqrt{-1}\over 2\pi} {\overset {0} g}_{\alpha\overline\beta} dz^\alpha\wedge d\overline z^\beta\) and \(H_{\Omega_0}\) be the space of all Kähler metrics with the same fixed \(\Omega_0\). Then, for any metric \(g\in H_{\Omega_0}\), there exists a real-valued scalar function \(\varphi\), globally defined on \(M\), such that: \(g_{\alpha\overline\beta}={\overset {0} g}_{\alpha\overline\beta}+ \varphi_{\alpha\overline\beta}\), where \(\varphi_{\alpha\overline\beta}= \partial^2\varphi/\partial z^\alpha\partial\overline z^\beta\). Let NEWLINE\[NEWLINE\begin{cases} {\partial F\over\partial t}= \Delta R= -\Delta^2F+\Delta (g^{\alpha\overline\beta}{\overset {0} R}_{\alpha\overline\beta}),\\ g_{\alpha\overline\beta}(z,\overline z,t)={\overset {0} g}_{\alpha\overline\beta}(z, \overline z)+ \varphi_{\alpha\overline\beta}(z,\overline z,t),\;t\geq 0,\\ \int_M e^{F_0}d\mu_0= \int_M d\mu_0,\;F_0(z,z)= F(z,\overline z,0)\end{cases}NEWLINE\]NEWLINE be the so-called modified Calabi flow on \((M,[\omega_0])\). (Here \(F: M\times[0,\infty)\to\mathbb{R}\) is a smooth function, \(\Delta= \Delta_{g_{\alpha\overline\beta}}\), \(d\mu_0= d\mu_{\overset {0} g}\), and \(d\mu= e^F d\mu_0\).)NEWLINENEWLINENEWLINEIn this paper, the author investigates the properties of this flow for \(m=2\). So, first, some \(C^\alpha\)-compactness properties for \(F\) are shown and applied to the Calabi flow. Some kind of Harnack estimate for the Calabi flow is proved. As consequences, under the following two conditions, the long-time existence and asymptotic convergence with no nonzero holomorphic tangent vector fields and \(R_0= -\Delta\log\text{det}({\overset {0} g}_{\alpha\beta})< 0\), is proved:NEWLINENEWLINENEWLINEi) \(F^i(z,\overline z)\) satisfies the so-called property \((*)\), i.e. there is a point \(x\in M\) and there exist positive constants \(\rho\), \(\varepsilon\), \(H\) independent of \(F^i\) such that \(\int_{B(x,\rho)} e^{-F^i} d\mu_0\leq H\).NEWLINENEWLINENEWLINEii) For fixed positive constants \(K\) and \(p\), \(\int_M|\text{Ric}(g^i)|^p d\mu_{g^i}\leq K\), \(p> 2\).NEWLINENEWLINENEWLINEHere \(\text{Ric}(g^i)\) is the Ricci curvature tensor with respect to \(g^i\).NEWLINENEWLINENEWLINEFinally, combining with the results from \textit{C. LeBrun} [Math. Res. Lett. 2, 653-662 (1995; Zbl 0874.53051)], the blow-up behavior of the Calabi flow on the ruled surface is shown.