
4. Geometric flows

1.     The Chern-Ricci curvature of a Hermitian metric $g$ is defined by $R^{\mathcal{Ch}}_{\overline k j} = -\partial_j \partial_{\overline k} \mathrm{log det}(g).$ A family of Hermitian metrics satisfies the Chern-Ricci flow if $\frac \partial {\partial t} g_{\overline k j} = -R^{\mathcal{Ch}}_{\overline k j}.$

Problem 4.1.

[Ben Weinkove] What is the behavior of the Chern-Ricci flow on the simplest Hopf surface, $\mathbb (\mathbb C^2 \setminus \{ 0\})/((z_1,z_2) \sim (2z_1,2z_2))$?
1. Remark. [Valentino Tosatti] Chern-Ricci flow is equivalent to a parabolic scalar PDE $\dot \varphi = \mathrm{log}\frac {\mathrm{det}(g_{\overline k j} - t R^{\mathcal{Ch}}_{\overline k j}+\varphi_{\overline k j})}{\mathrm{det}(g_{\overline k j})}$ for $g_{\overline k j}$ initial Hermitian metric.
• Problem 4.2.

[Gábor Székelyhidi] What is the behavior of $g_{j \overline k}(t)$ solving the Calabi flow, $\frac \partial {\partial t} g_{j \overline k} = \partial_j \partial_{\overline k} R?$ Can long time existence be established for toric surfaces?
• Problem 4.3.

[Yuri Ustinovskiy] Consider the flow of inverse Hermitian metrics on a compact complex manifold: $\frac \partial {\partial t} g^{j \overline k} = g^{m \overline n} \partial_m \partial_{\overline n} g^{j \overline k} - \partial_m g^{j \overline k} \partial_{\overline n} g^{m \overline k}.$ One has short time existence, and it is known that at the maximal time of existence $| \mathrm{Rm}^{\mathcal{Ch}} | + |T| + |\nabla T| \to \infty.$ Can this be improved? Can one find analogues of Perelman’s $\mathcal F$ and $\mathcal W$ functionals?
• Problem 4.4.

[Tristan Collins] Is the diameter bounded at finite time singularities of the Kähler-Ricci flow on compact Kähler manifolds?

Cite this as: AimPL: Nonlinear PDEs in real and complex geometry, available at http://aimpl.org/nonlinpdegeom.