[Paper Review] Recent progress on the Yamabe problem

Simon Brendle, Fernando C. Marques|arXiv (Cornell University)|Oct 24, 2010

Nonlinear Partial Differential Equations33 references20 citations

TL;DR

This paper surveys recent advances in the Yamabe problem, focusing on compactness and non-compactness results for solutions to the Yamabe equation. It establishes the convergence of the Yamabe flow to a constant scalar curvature metric under conditions such as local conformal flatness or the absence of certain Weyl tensor singularities, resolving long-standing conjectures in dimensions 3–5 and higher under geometric constraints.

ABSTRACT

We give a survey of various compactness and non-compactness results for the Yamabe equation. We also discuss a conjecture of Hamilton concerning the asymptotic behavior of the parabolic Yamabe flow.

Motivation & Objective

• To survey recent developments in the Yamabe problem, particularly regarding the existence and compactness of solutions to the Yamabe equation.
• To investigate the asymptotic behavior of the parabolic Yamabe flow, especially under geometric constraints.
• To resolve the Compactness Conjecture in specific cases by proving uniform bounds and convergence of the Yamabe flow.
• To extend the convergence results of the Yamabe flow to higher dimensions (n ≥ 6) under conditions involving the Weyl tensor and spin structure.

Proposed method

• Utilizes the variational characterization of the Yamabe functional $ E_{g_0}(u) $, minimizing it over positive smooth functions to find solutions to the Yamabe equation.
• Applies the Positive Mass Theorem in dimensions 3–5 to construct test functions with energy below $ Y(S^n) $, ensuring existence of minimizers.
• Employs a parabolic flow approach via the Yamabe flow $ g(t) $, proving convergence to a constant scalar curvature metric by controlling $ L^2 $-norms of scalar curvature fluctuations.
• Introduces a refined energy estimate involving $ \int_0^\infty \|R_{g(\tau)} - r_{g(\tau)}\|_{L^2} \, d\tau < \infty $ to rule out volume concentration.
• Constructs test functions using Aubin’s method generalized to higher dimensions, ensuring $ E_{g_0}(u) < Y(S^n) $ when $ \mathcal{Z} = \emptyset $ or $ M $ is spin.
• Uses ODE comparison lemmas to derive decay estimates on scalar curvature fluctuations, enabling uniform bounds on the conformal factor $ u(t) $.

Experimental results

Research questions

• RQ1 Under what geometric conditions does the Yamabe flow converge to a metric of constant scalar curvature?
• RQ2 Does the set of solutions to the Yamabe equation remain compact under the Palais-Smale condition failure?
• RQ3 Can the convergence of the Yamabe flow be established in dimensions $ n \geq 6 $ when the manifold is not conformally flat?
• RQ4 What role does the Weyl tensor play in the formation of singularities or non-compactness in the solution space?
• RQ5 How do the assumptions of local conformal flatness or spin structure affect the long-term behavior of the Yamabe flow?

Key findings

• The Yamabe flow converges to a constant scalar curvature metric for compact manifolds of dimension $ 3 \leq n \leq 5 $, provided the initial metric is not conformally equivalent to the standard sphere.
• For $ n \geq 6 $, the Yamabe flow converges to a constant scalar curvature metric if $ \mathcal{Z} = \emptyset $ or $ M $ is spin, where $ \mathcal{Z} $ is the set of points where the Weyl tensor decays sufficiently fast.
• The $ L^2 $-norm of the scalar curvature fluctuation $ R_{g(t)} - r_{g(t)} $ decays over time, with $ \int_0^\infty \|R_{g(\tau)} - r_{g(\tau)}\|_{L^2} \, d\tau < \infty $, implying uniform control on the conformal factor.
• Volume concentration is ruled out via a uniform bound on geodesic ball volumes, ensuring $ \text{vol}(B_r(p), g(t)) \leq \eta $ for all $ t \geq 0 $ and $ p \in M $, for some $ r > 0 $.
• The solution to the Yamabe problem is unique up to scaling if $ Y(M, g_0) \leq 0 $, but non-uniqueness arises when $ Y(M, g_0) > 0 $, with arbitrarily many solutions possible under small perturbations.
• The proof of convergence relies on constructing test functions with energy below $ Y(S^n) $, using the Weyl tensor in non-flat cases and the Positive Mass Theorem in low dimensions.