The Chern-Ricci flow on complex surfaces (Q2873600)

The authors consider the Chern-Ricci flow, a flow of Hermitian metrics, introduced by \textit{M. Gill} [Commun. Anal. Geom. 19, No. 2, 277--304 (2011; Zbl 1251.32035)], and given on complex manifolds by the evolution equation NEWLINE\[NEWLINE\partial_t \omega = -\mathrm{Ric}(\omega)\tag{CRF}NEWLINE\]NEWLINE for \(t\geq 0\), where, in local coordinates, \({\text{ Ric}}(\omega):=-i\partial\bar{\partial}\log\det(g_{a\bar{b}})\) is the \textit{Chern-Ricci form} of \(\omega=g_{a\bar{b}} idz^a\wedge d\overline{z^b}\). If the initial metric \(\omega_0\) is Gauduchon (i.e. if \(i\partial\bar{\partial}\omega_0=0\)) -- a condition preserved by the flow -- the authors already proved existence and uniqueness of the flow on a maximal interval \([0,T)\) for some \(T=T(\omega_0)\in(0,\infty]\) in [``On the evolution equation of a Hermitian metric by its Chern-Ricci flow'', Preprint, \url{arXiv:1201.0312}]. NEWLINENEWLINENEWLINEIn continuity with this previous work, and mostly restricting to the case of complex surfaces, the authors give in the first part of the present article two general convergence results for \(T<\infty\), and the volume of the surface under study stays positively bounded along the flow (`\textit{finite time non-collapsing}' case, holding for example on surfaces of nonnegative Kodaira dimension). They show that, first, on the complement of finitely many disjoint \((-1)\)-curves, \((\omega_t)\) converges smoothly to a (smooth) metric \(\omega_T\) (Theorem 1.1); and secondly, calling \(M\) the surface and \(N\) the surface with contracted \((-1)\)-curves, and under an additional condition \((*)\) -- easily verified for a good choice of \(\omega_0\) --, that there exists a distance \(d_T\) on \(N\) such that \((M,\omega_t)\) converges in the Gromov-Hausdorff sense to \((N,d_T)\) (Theorem 1.3).NEWLINENEWLINENEWLINEProofs of these statements are given respectively in \S2 and \S3 of the paper. Besides deep structure results on Kähler and non-Kähler surfaces the proof of Theorem 1.1 in \S2 is based on successive a priori estimates on the (normalised) function \(\varphi=\varphi_t\) (verifying \(\omega_t=\omega_0+i\partial\bar{\partial}\varphi_t\)), and on a log-corrected version \(\tilde{\varphi}\) (going to \(+\infty\) near the \((-1)\)-curves mentioned above): upper bounds on \(\varphi\), lower bounds on \(\tilde{\varphi}\), and so on. The classical maximum principle is applied to get the desired estimates, together with another ingredient for the second-order estimates, consisting in a clever use of the term \(\frac{1}{\tilde{\varphi}+C}\) (with \(C\) such that \(\tilde{\varphi}+C\geq 1\)), a trick due to \textit{D. H. Phong} and \textit{J. Sturm} [Commun. Anal. Geom. 18, No. 1, 145--170 (2010; Zbl 1222.32044)].NEWLINENEWLINENEWLINEIn \S3, the additional condition \((*)\) is used to get some estimates missing in the previous paragraph (e.g. an upper bound on \(\varphi\)); Phong-Sturm's trick is now moreover used in an essential way to deal with torsion terms (by comparison with the Kähler setting) in the technical core of the paragraph, namely Lemmas 3.4 and 3.5. The end of \S3 is devoted to a precise discussion on a comparison of Theorems 1.1 and 1.3, with their analogues for the Kähler-Ricci flow on surfaces; it is particularly emphasised that \((N,d_T)\) being the metric completion of \(M\backslash\{(-1)-\text{ curves}\}\) endowed with \(g_T\) is still a very delicate problem in the Chern-Ricci flow framework.NEWLINENEWLINENEWLINEA second part of the paper, \(\S\S\)4--8, is devoted to another type of convergence results. Namely, in these paragraphs, the authors examine carefully some special explicit examples of \textit{collapsing} Chern-Ricci flows on various classes of surfaces, either in finite time \((T<\infty)\) or infinite-time \(T=\infty\); the infinite-time collapsing flows are normalized Chern-Ricci flows though: the volume going to 0 as \(t\to\infty\) is that of \(\frac{\omega(t)}{t}\). Let us synthesize their observations:NEWLINENEWLINENEWLINE1) In \(\S\)4, Hopf surfaces \(H=(\mathbb{C}^2\backslash\{0\})/_\sim\) are considered, where \((z_1,z_2)\sim(\alpha_1z_1,\alpha_2z_2)\), \(|\alpha_1| =|\alpha_2|\neq 1\), with initial metric \(\omega_0=\frac{\delta_{ab}}{r^2}idz^a\wedge d\overline{z^b}\). Then \(T=\frac{1}{2}\), \(\omega(t)\) converges to a nonnegative (but not positive) smooth \((1,1)\)-form, and \((H,\omega(t))\) converges to the usual metric circle (with radius related to \(|\alpha_1|\)) in the Gromov-Hausdorff sense. This holds actually in any dimension.NEWLINENEWLINENEWLINE2) In \(\S\S\)5--8 the three families of Inoue surfaces are considered. These surfaces can be written as \((H\times\mathbb C)/\Gamma\), where \(H\) is the Poincaré half-plane and \(\Gamma\) a subgroup of \({\mathrm{ Aut}}(H\times\mathbb C)\). Here \(T=\infty\), and the normalized Chern-Ricci flow, starting at Gauduchon metrics discovered by \textit{F. Tricerri} [Rend. Semin. Mat., Torino 40, No. 1, 81--92 (1982; Zbl 0511.53068)], collapses; for the first family, the flow contracts \(\mathbb T^3\)-fibres, and for the second family -- to which the third family case is also reduced by taking a double cover --, more general 3-dimensional fibres are contracted by the flow. In these cases, the limit \(\frac{\omega(t)}{t}\) tends to the pull-back of the Poincaré metric, whereas \((M,\frac{\omega(t)}{t})\) converges again to a standard metric circle (with radius parametrised by \(\Gamma\)) in the Gromov-Hausdorff sense.NEWLINENEWLINENEWLINE3) The last class of examples are non-Kähler properly elliptic surfaces, i.e. surfaces \(S\) with \(b_1\) odd, of Kodaira dimension 1, and admitting an elliptic fibration \(\pi:S\to C\) (\(C\): smooth compact curve). The initial Gauduchon metric is here the one discovered by \textit{I. Vaisman} [Rend. Semin. Mat., Torino 45, No. 3, 117--123 (1987; Zbl 0696.53039)]; again \(T=\infty\), and \(\frac{\omega(t)}{t}\) tends to \(\pi^*\omega_{\text{KE}}\) whereas \((S,\frac{\omega(t)}{t})\) converges Gromov-Hausdorff to \((C,d_{\text{KE}})\); here \(\omega_{\text{KE}}\) is an orbifold Kähler-Einstein metric (with \(\mathrm{Ric}=-1\)), inducing \(d_{\text{KE}}\), on \(C\).NEWLINENEWLINENEWLINEFinally, in the conclusive \S9, the authors generalise the \textit{Mabuchi energy} of Kähler geometry to the setting of complex surfaces with vanishing first Bott-Chern class. They show that this functional is decreasing along the Chern-Ricci flow, and discuss how this can be used to reprove a result by Gill -- when the first Bott-Chern class is 0, the Chern-Ricci flow converges smoothly to a Chern-Ricci flat metric -- in the special case of Gauduchon initial metric and complex dimension 2.