On Critical Dimensions for Compactness in the Boundary Yamabe Problem, II
Abstract
We determine the sharp compactness ranges for the scalar-flat and minimal-boundary Yamabe problems on smooth compact manifolds of positive conformal type, excluding the conformal round hemisphere. For zero scalar curvature and positive constant boundary mean curvature, compactness holds through dimension $14$ for general boundary and dimension $21$ for umbilic boundary. For positive scalar curvature and zero boundary mean curvature, the corresponding upper dimensions are $14$ and $20$. Together with the noncompactness examples in Part I, these results identify the transition dimensions in both boundary classes. For positive scalar curvature, we also prove compactness through dimension eight for every fixed real boundary mean curvature, and obtain higher-dimensional ranges when this curvature is near zero or sufficiently large and positive. The proof combines scalar-correction estimates for the full conformal Fermi metric expansion with a geometric formula expressing the logarithmic coefficient of the corrected energy as a negative sum of squares.
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BibTeX
@misc{gong2026critical,
title = {{On Critical Dimensions for Compactness in the Boundary Yamabe Problem, II}},
author = {Liuwei Gong and Seunghyeok Kim and Monica Musso and Juncheng Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10485},
eprint = {2610.10485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10485},
}
RIS
TY - UNPB
TI - On Critical Dimensions for Compactness in the Boundary Yamabe Problem, II
AU - Liuwei Gong
AU - Seunghyeok Kim
AU - Monica Musso
AU - Juncheng Wei
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.10485
UR - https://arxiv.org/abs/2610.10485
AB - We determine the sharp compactness ranges for the scalar-flat and minimal-boundary Yamabe problems on smooth compact manifolds of positive conformal type, excluding the conformal round hemisphere. For zero scalar curvature and positive constant boundary mean curvature, compactness holds through dimension $14$ for general boundary and dimension $21$ for umbilic boundary. For positive scalar curvature and zero boundary mean curvature, the corresponding upper dimensions are $14$ and $20$. Together with the noncompactness examples in Part I, these results identify the transition dimensions in both boundary classes. For positive scalar curvature, we also prove compactness through dimension eight for every fixed real boundary mean curvature, and obtain higher-dimensional ranges when this curvature is near zero or sufficiently large and positive. The proof combines scalar-correction estimates for the full conformal Fermi metric expansion with a geometric formula expressing the logarithmic coefficient of the corrected energy as a negative sum of squares.
ER -