4. Geometric flows
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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, $\mathbb (\mathbb C^2 \setminus \{ 0\})/((z_1,z_2) \sim (2z_1,2z_2))$?-
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.
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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.