Asymptotic behavior of solutions to the Yamabe equation with an asymptotically flat metric

TL;DR

This paper characterizes the asymptotic behavior of positive solutions to the Yamabe equation on asymptotically flat manifolds, showing they converge to fundamental or Fowler solutions under certain flatness and dimension conditions.

Contribution

It extends previous results by establishing asymptotic behaviors for solutions to the Yamabe equation in higher dimensions and under minimal flatness conditions, including cases with non-constant scalar curvature.

Findings

Solutions converge to fundamental or Fowler solutions at infinity.

Results hold for dimensions up to 24 and for solutions near singularities.

Established asymptotic behavior under minimal flatness conditions.

Abstract

We prove that any positive solution of the Yamabe equation on an asymptotically flat $n$-dimensional manifold of flatness order at least $\frac{n-2}{2}$ and $n\le 24$ must converge at infinity either to a fundamental solution of the Laplace operator on the Euclidean space or to a radial Fowler solution defined on the entire Euclidean space. The flatness order $\frac{n-2}{2}$ is the minimal flatness order required to define ADM mass in general relativity; the dimension $24$ is the dividing dimension of the validity of compactness of solutions to the Yamabe problem. We also prove such alternatives for bounded solutions when $n>24$. We prove these results by establishing appropriate asymptotic behavior near an isolated singularity of solutions to the Yamabe equation when the metric has a flatness order of at least $\frac{n-2}{2}$ at the singularity and $n\le 24$, also when $n>24$ and the solution grows no faster than the fundamental solution of the flat metric Laplacian at the singularity. These results extend earlier results of L. Caffarelli, B. Gidas and J. Spruck, also of N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, when the metric is conformally flat, and work of C.C. Chen and C. S. Lin when the scalar curvature is a non-constant function with appropriate flatness at the singular point, also work of F. Marques when the metric is not necessarily conformally flat but smooth, and the dimension of the manifold is three, four, or five, as well as recent similar results by the second and third authors in dimension six.