Kähler-Ricci flow on homogeneous toric bundles

Hong Huang

Paper · May 22, 2017

Kähler-Ricci flow on homogeneous toric bundles

Hong Huang

math.DGarXiv:1705.07735v6

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Abstract

Assume that $X$ is a homogeneous toric bundle of the form $G^{C} \times_{P, τ} F$ and is Fano, where $G$ is a compact semisimple Lie group with complexification $G^{C}$ , $P$ a parabolic subgroup of $G^{C}$ , $τ : P \to (T^{m})^{C}$ is a surjective homomorphism from $P$ to the algebraic torus $(T^{m})^{C}$ , and $F$ is a compact toric manifold of complex dimension $m$ . In this note we show that the normalized Kähler-Ricci flow on $X$ with a $G \times T^{m}$ -invariant initial Kähler form in $c_{1} (X)$ converges, modulo the algebraic torus action, to a Kähler-Ricci soliton. This extends a previous work of X. H. Zhu. As a consequence we recover a result of Podestà-Spiro.

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Submitted: May 22, 2017
Last revised: Dec 30, 2019 (v6)
Subjects: math.DG, math.AP
DOI: 10.1142/S0129167X20500226

Comments: to appear in International Journal of Mathematics

Ricci flow Kähler–Ricci flow Ricci solitons Kähler–Ricci solitons Kähler geometry Homogeneous spaces

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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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000)

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.
11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.
11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

[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] 11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.

[3] 11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

Kähler-Ricci flow on homogeneous toric bundles

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