Uniqueness of shrinking gradient Kähler-Ricci solitons on non-compact toric manifolds

Charles Cifarelli

Paper · Oct 1, 2020

Uniqueness of shrinking gradient Kähler-Ricci solitons on non-compact toric manifolds

Charles Cifarelli

math.DGarXiv:2010.00166v3

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Abstract

We show that, up to biholomorphism, there is at most one complete $T^{n}$ -invariant shrinking gradient Kähler-Ricci soliton on a non-compact toric manifold $M$ . We also establish uniqueness without assuming $T^{n}$ -invariance if the Ricci curvature is bounded and if the soliton vector field lies in the Lie algebra $t$ of $T^{n}$ . As an application, we show that, up to isometry, the unique complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on $CP^{1} \times C$ is the standard product metric associated to the Fubini-Study metric on $CP^{1}$ and the Euclidean metric on $C$ .

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Submitted: Oct 1, 2020
Last revised: Jul 18, 2022 (v3)
Subjects: math.DG, math.AG, math.SG

Comments: accepted version, to appear in J. London Math. Soc

Ricci solitons Kähler–Ricci solitons Kähler geometry Scalar curvature Stability & uniqueness Boundaries & non-compact

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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 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) 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.
11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

[1] 6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

[2] 9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

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

Uniqueness of shrinking gradient Kähler-Ricci solitons on non-compact toric manifolds

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