Comparison of the Calabi and Mabuchi geometries and applications to geometric flows

Tamás Darvas

Paper · Feb 13, 2016

Comparison of the Calabi and Mabuchi geometries and applications to geometric flows

Tamás Darvas

math.DGarXiv:1602.04309v2

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Abstract

Suppose $(X, ω)$ is a compact Kähler manifold. We introduce and explore the metric geometry of the $L^{p, q}$ -Calabi Finsler structure on the space of Kähler metrics $H$ . After noticing that the $L^{p, q}$ -Calabi and $L^{p^{'}}$ -Mabuchi path length topologies on $H$ do not typically dominate each other, we focus on the finite entropy space $E^{E n t}$ , contained in the intersection of the $L^{p}$ -Calabi and $L^{1}$ -Mabuchi completions of $H$ and find that after a natural strengthening, the $L^{p}$ -Calabi and $L^{1}$ -Mabuchi topologies coincide on $E^{E n t}$ . As applications to our results, we give new convergence results for the Kähler–Ricci flow and the weak Calabi flow.

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Submitted: Feb 13, 2016
Last revised: Aug 13, 2016 (v2)
Subjects: math.DG, math.CV, math.MG
Journal: Ann. I.H.P. Anal. Non Lineaire 34 (2017), no. 5, 1131-1140

Comments: v2 Final version. References added

Ricci flow Kähler–Ricci flow Other curvature flows Kähler geometry Kähler–Einstein & K-stability

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.
5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

[1] 5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.

[2] 5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

[3] 10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

Comparison of the Calabi and Mabuchi geometries and applications to geometric flows

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