The Calabi flow on Kähler surfaces with bounded Sobolev constant. I
This is another effort on the Calabi flow on toric surfaces. In complex geometry, a central problem is to find Kähler-Einstein metrics on Fano manifolds, i.e., compact Kähler manifolds with positive Ricci classes. For a general Kähler class on a general compact Kähler manifold, there is the similar question of finding Kähler metrics with constant scalar curvatures. We notice that any Kähler metric with constant scalar curvature in the Ricci class is automatically Kähler-Einstein. Historically, there are several methods to obtain this kind of metrics. 1. The Calabi-Aubin-Yau continuity method (see [\textit{Y.-T. Siu}, Ann. Math. (2) 127, No. 3, 585--627 (1988; Zbl 0651.53035); \textit{G. Tian}, Invent. Math. 89, 225--246 (1987; Zbl 0599.53046)]). They use \(\operatorname{Ric}(\omega _t )=t\omega +(1-t)\operatorname{Ric} (\omega _0 )\) with \(0\leq t\leq 1\). 2. The Ricci flow (see [\textit{G. Tian} and \textit{X. Zhu}, J. Am. Math. Soc. 20, No. 3, 675--699 (2007; Zbl 1185.53078)]). 3. The Calabi flow. 4. The iterating inverse Ricci operation, which the reviewer discussed with Tian on several occasions (see [\textit{Y. A. Rubinstein}, Adv. Math. 218, No. 5, 1526--1565 (2008; Zbl 1143.53065)]). More recently, Kähler-Einstein metrics (or some special metrics) on open manifolds were used to approach Kähler-Einstein metrics (see [the reviewer and the first author, Asian J. Math. 4, No. 4, 817--830 (2000; Zbl 1003.32003); \textit{C. Arezzo} and \textit{F. Pacard}, Acta Math. 196, No. 2, 179--228 (2006; Zbl 1123.53036)]; see also [the first author, \textit{C. LeBrun} and \textit{B. Weber}, J. Am. Math. Soc. 21, No. 4, 1137--1168 (2008; Zbl 1208.53072)]). For a general Kähler class, there are some special metrics generalizing the Kähler metrics with constant scalar curvature. For example, if the scalar curvature is a potential function of a holomorphic vector field, then we call the Kähler metric a Calabi extremal metric (see [\textit{E. Calabi}, Ann. Math. Stud. 102, 259--290 (1982; Zbl 0487.53057)]). There is a Calabi flow \[ \dot{\varphi }=\operatorname{R}-\operatorname{HR}, \] where \(\operatorname{R}\) is the scalar curvature and \(\operatorname{HR}\) is the average of the scalar curvature. Also, there is another standard metric given in [the reviewer, Int. J. Math. 6, No. 3, 371--379 (1995; Zbl 0847.53032)] with a modified Ricci flow \[ \dot{g}=-\operatorname{Ric} +\operatorname{HRic}, \] where \(\operatorname{Ric}\) is the Ricci curvature and \(\operatorname{HRic}\) the harmonic part of the Ricci curvature. We notice that, when \(g\) is in the Ricci class, this is just the normalized Ricci flow. In 1991, \textit{P. T. Chruściel} gave a surprising proof in [Commun. Math. Phys. 137, No. 2, 289--313 (1991; Zbl 0729.53071)] that the Calabi flow converges to metrics with constant curvatures on the Riemann surfaces. I talked the matter over with P. Yang in 1996 when I talked about the modified Ricci flow. He said to let a young person work on it. So, I just waited. The first author was a student of Calabi working on generalizations of Chruściel's work, who showed me [Chruściel, loc. cit.] eventually in Princeton 1999. Later on, \textit{S.-C. Chang}, in [J. Math. Kyoto Univ. 40, No. 2, 363--377 (2000; Zbl 1098.53505); Nagoya Math. J. 181, 63--73 (2006; Zbl 1098.53031)], proved that the Calabi flow converges on compact Riemann surfaces without assuming the existence of the Einstein metric. The reviewer proved the first higher-dimensional convergence result of Calabi flow with symmetries in [Pac. J. Math. 233, No. 1, 91--124 (2007; Zbl 1154.53043)] and told this Tian in England 2001. In that paper, it turned out that the Calabi flow is easier than the modified Ricci flow for the hypersurface-ends cases. However, the long time existence might not be true even for \(\mathbb CP^n\) with \(n>2\) (with higher codimensional ends). In this paper, the authors generalize the calculations in the proof of the Lemma 9.3 in the reviewer's paper mentioned above to bound the higher-order derivatives of the curvature and obtains a Donaldson type of bubbling [\textit{S. K. Donaldson}, J. Differ. Geom. 79, No. 3, 389--432 (2008; Zbl 1151.53030)] for complex surfaces. Compared to Donaldson's paper, their proof is a little bit sketchy. Then, they prove small energy convergence for the Calabi flow on the blow-up of \(\mathbb CP^2\) at two points using the Yamabi constant estimate as in [the first author, LeBrun and Weber, loc. cit.]. Notice that the bubblings appearing in this paper differ from the bubblings appearing in the paper of the reviewer with the first author [loc. cit.] and in the paper of the reviewer [Int. J. Math. 14, No. 3, 259--287 (2003; Zbl 1048.32014)]. The Donaldson type of bubblings are scalar flat and are difficult to get rid of. One has to check case by case and for different original manifolds to use different arguments. The bubblings that appeared in our papers are Kähler-Einstein metrics on the (open) normal bundles over a Kähler-Einstein submanifolds and can be treated by an integral uniformly by the positive lemma result in [the reviewer, Trans. Am. Math. Soc. 347, No. 6, 2255--2262 (1995; Zbl 0853.53047)]; see also [the reviewer, Pac. J. Math. 253, No. 2, 383--422 (2011; Zbl 1241.32020)].
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