Curvature estimates for steady Ricci solitons

Pak-Yeung Chan

Paper · Aug 27, 2019

Curvature estimates for steady Ricci solitons

Pak-Yeung Chan

math.DGarXiv:1908.10456

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Abstract

We show that for an $n$ dimensional complete non Ricci flat gradient steady Ricci soliton with potential function $f$ bounded above by a constant and curvature tensor $R m$ satisfying $lim_{r \to \infty} r ∣ R m ∣ < \frac{1}{5}$ , then $∣ R m ∣ \leq C e^{- r}$ for some constant $C > 0$ , improving a result of [36]. For any four dimensional complete non Ricci flat gradient steady Ricci soliton with scalar curvature $S \to 0$ as $r \to \infty$ , we prove that $∣ R m ∣ \leq c S$ for some constant $c > 0$ , improving an estimate in [11]. As an application, we show that for a four dimensional complete non Ricci flat gradient steady Ricci soliton, $∣ R m ∣$ decays exponentially provided that $lim_{r \to \infty} r S$ is sufficiently small and $f$ is bounded above by a constant.

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Submitted: Aug 27, 2019
Subjects: math.DG

Ricci solitons Ricci-flat & Calabi–Yau 4-manifolds Scalar curvature

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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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 1982) 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.

Curvature estimates for steady Ricci solitons

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