TY - JOUR
T1 - Asymptotic behavior of solutions to the Yamabe equation with an asymptotically flat metric
AU - Han, Zheng Chao
AU - Xiong, Jingang
AU - Zhang, Lei
N1 - Publisher Copyright:
© 2023 Elsevier Inc.
PY - 2023/8/15
Y1 - 2023/8/15
N2 - We prove that any positive solution of the Yamabe equation on an asymptotically flat n-dimensional manifold of flatness order at least [Formula presented] and n≤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 [Formula presented] 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 [Formula presented] at the singularity and n≤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.
AB - We prove that any positive solution of the Yamabe equation on an asymptotically flat n-dimensional manifold of flatness order at least [Formula presented] and n≤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 [Formula presented] 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 [Formula presented] at the singularity and n≤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.
KW - Asymptotic behavior
KW - Asymptotically flat metric
KW - Isolated singularity
KW - Yamabe equation
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U2 - 10.1016/j.jfa.2023.109982
DO - 10.1016/j.jfa.2023.109982
M3 - Article
AN - SCOPUS:85156196263
SN - 0022-1236
VL - 285
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 4
M1 - 109982
ER -