[Paper Review] Isolated singularities of solutions to the Yamabe equation in dimension $6$

Jingang Xiong, Lei Zhang|arXiv (Cornell University)|Jun 23, 2020

Nonlinear Partial Differential Equations27 references4 citations

TL;DR

This paper establishes the asymptotic behavior of positive solutions to the Yamabe equation near an isolated singularity in dimension 6, proving that such solutions are asymptotically close to a Fowler solution with an error term of order $ O(|x|^{\alpha}) $ for some $ \alpha > 0 $. The result extends Marques' earlier work from dimensions 3–5 to the borderline case of dimension 6, using refined analysis of the conformal Laplacian and Pohozaev-type identities under non-conformally flat metrics.

ABSTRACT

We study the asymptotic behavior of local solutions to the Yamabe equation near an isolated singularity, when the metric is not conformally flat. We prove that, in dimension $6$, any solution is asymptotically close to a Fowler solution, which is an extension of the same result for lower dimensions by F.C. Marques in 2008.

Motivation & Objective

• To extend the asymptotic behavior result for isolated singularities of the Yamabe equation from dimensions 3–5 to dimension 6.
• To address the challenge that the metric is not conformally flat, which breaks symmetry and complicates analysis.
• To establish sharp upper and lower bounds on the growth of solutions near the singularity using conformal geometry and maximum principle techniques.
• To prove that solutions in dimension 6 are asymptotically close to Fowler solutions, confirming the validity of the method in the critical dimension.

Proposed method

• Use of the moving spheres method with carefully constructed test functions to derive an upper bound on the solution growth.
• Application of conformal normal coordinates to handle non-conformally flat metrics and localize the analysis.
• Employment of Pohozaev-type identities and integral criteria to characterize removable singularities.
• Deformation of the metric conformally to one with negative scalar curvature to exploit monotonicity properties in the lower bound estimate.
• Use of radial transformation and weighted energy estimates via $ w(t) = u(e^{-t})e^{(n-2)t/2} $ to analyze decay rates.
• Combination of differential inequalities and asymptotic analysis to control error terms and prove $ u(x) = u_0(1 + O(|x|^{\alpha})) $.

Experimental results

Research questions

• RQ1 Does the asymptotic behavior of solutions to the Yamabe equation near an isolated singularity persist in dimension 6 when the metric is not conformally flat?
• RQ2 Can the upper and lower bounds on solution growth be established in the borderline case $ n = 6 $, where the standard method fails due to $ \tau = 2 = \frac{n-2}{2} $?
• RQ3 Is the Fowler solution still the asymptotic model for singular solutions in dimension 6, despite the loss of symmetry?
• RQ4 Can the $ O(|x|^{\alpha}) $ error estimate be achieved in dimension 6, as in lower dimensions?
• RQ5 What role does the Pohozaev integral play in determining the removability of the singularity in this critical dimension?

Key findings

• The upper bound $ \limsup_{x\to 0} d_g(x,0)^{2} u(x) < \infty $ holds in dimension 6, ensuring controlled growth near the singularity.
• The lower bound $ u(x) \geq \frac{1}{C} |x|^{-2} $ is established, confirming non-removable singularities.
• The Pohozaev integral vanishes for non-removable singularities, implying that the singularity is not removable if the solution does not decay too fast.
• The solution satisfies $ u(x) = u_0(x)(1 + O(|x|^{\alpha})) $ as $ x \to 0 $, where $ u_0 $ is a Fowler solution and $ \alpha > 0 $, extending Marques' result to dimension 6.
• The proof relies on delicate analysis of the conformal Laplacian and refined estimates in conformal normal coordinates, particularly in the critical case $ n = 6 $.
• The result confirms that the method of Marques fails in dimension 6 unless the metric flatness condition $ \tau > \frac{n-2}{2} $ is strengthened, which is not satisfied here.