Metric contraction of the cone divisor by the conical Kähler–Ricci Flow

Abstract

We use the momentum construction of Calabi to study the conical Kähler-Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov-Hausdorff converges to the Riemann sphere or a single point in finite time, or the flow contracts the cone divisor to a single point and Gromov-Hausdorff converges to a two dimensional projective orbifold. This gives the first example of the conical Kähler-Ricci flow contracting the cone divisor to a single point. At the end, we introduce a conjectural picture of the geometry of finite time non-collapsing singularities of the flow on Kähler surfaces in general.

Conical Kähler–Ricci flows on Fano manifolds

“…These flows were first proposed in Jeffres-Mazzeo-Rubinstein's paper (see Section 2.5 in [26]), then Song-Wang (conjecture 5.2 in [48]) made a conjecture on the relation between the convergence of these flows and the greatest Ricci lower bounds of the manifolds. Then the existence, regularity and limit behavior of the conical Kähler-Ricci flows have been studied by Chen-Wang [9,10], Edwards [19,20], Liu-Zhang [34], Liu-Zhang [36,37], Nomura [39], Shen [45,46], Wang [57] and Zhang [62,63] etc.…”

On a twisted conical Kähler–Ricci flow

“…We point out that the first result that the conical Kähler-Ricci flow will collapse the cone divisor to a point in Gromov-Hausdorff topology was recently obtained by Edwards in [7] on Hirzebruch surfaces (in which Edwards also proved that, in some finite-time noncollapsing case, the conical Kähler-Ricci flow can contract cone divisor to a point in Gromov-Hausdorff topology), under certain symmetry condition on the initial model conical metric. Our Theorem 1.2 (2.3) provides another result on such phenomenon.…”

“…The conical Kähler-Ricci flow is the Kähler-Ricci flow with certain cone singularities, whose existence, regularity and convergence have been widely studied in the recent years, see e.g. [4,5,6,7,17,18,21,33,40,43,45] and references therein.…”

The Anomaly Flow over Riemann Surfaces

“…The conservation laws presented here arise from the fact that the Anomaly flow preserves the conformally balanced cohomology class [ Ω ω ω 2 ] ∈ H 4 (X; R). In the case of generalized Calabi-Gray manifolds with our ansatz, the de Rham conformally balanced cohomology class is parameterized exactly by the vector V , as can be seen by the expression (7).…”

“…The limiting metric ωB which appears on the base B in the limit of the flow is a twisted Kähler-Einstein metric which solves Ric(ωB) = −ωB + ωW P , where ωW P is the Weil-Petersson metric of the fibration. Other flows in complex geometry, such as the Chern-Ricci flow and the conical Kähler-Ricci flow, also exhibit collapsing behavior [54,8,60,7].…”