Convergence of the Weighted Yamabe Flow

Zetian Yan

Paper · Nov 10, 2021

Convergence of the Weighted Yamabe Flow

Zetian Yan

math.DGWider flowsarXiv:2111.05945v2

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Abstract

We introduce the weighted Yamabe flow $\frac{\partial g}{\partial t} = (r_{φ}^{m} - R_{φ}^{m}) g$ , $\frac{\partial φ}{\partial t} = \frac{m}{2} (R_{φ}^{m} - r_{φ}^{m})$ on a smooth metric measure space $(M^{n}, g, e^{- φ} dvol_{g}, m)$ , where $R_{φ}^{m}$ denotes the associated weighted scalar curvature, and $r_{φ}^{m}$ denotes the mean value of the weighted scalar curvature. We prove long-time existence and convergence of the weighted Yamabe flow if the dimension $n$ satisfies $n ⩾ 3$ .

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Submitted: Nov 10, 2021
Last revised: Apr 14, 2023 (v2)
Subjects: math.DG

Comments: Add several estimates needed in the proof of Proposition 3.3; see Proposition 4.7, 4.8 and Lemma 4.10 for precise statement

Yamabe flow Scalar curvature Limit spaces & RCD Long-time existence & convergence

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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.

Yamabe flow Wider flows The Yamabe flow deforms a metric within its conformal class, driven by scalar curvature. It is a parabolic approach to the Yamabe problem: find a… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.
6C.11
Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.
9B.4
Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds
Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

[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] 6C.11
Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

[3] 9B.4
Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds
Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

Convergence of the Weighted Yamabe Flow

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