On critical dimensions for compactness in the boundary Yamabe problem, I
Abstract
We construct smooth, non-locally-conformally-flat metrics on the closed ball for which the boundary Yamabe equation admits L∞-unbounded sequences of positive solutions. The background metric and the prescribed scalar and boundary mean curvatures are fixed along each sequence. For zero scalar curvature and positive boundary mean curvature, such examples exist with umbilic boundary in every dimension N≥22 and with nonumbilic boundary in every N≥15. For positive scalar curvature and minimal boundary, the corresponding ranges are N≥21 and N≥15. For every N≥9, examples with either umbilic or nonumbilic boundary also exist when the scalar curvature is N(N−1) and the prescribed boundary mean curvature is below a negative threshold depending on N. The construction uses polynomial metric perturbations and corrected bubbles on the half-space, followed by conformal compactification to the ball. We evaluate or estimate the correction solving the linearized Neumann or Robin boundary problem and its contribution to the quadratic term of the reduced energy. We prove that this quadratic term has negative, nondegenerate local extrema with respect to tangential translations and scale in every stated dimension.
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BibTeX
@misc{gong2026critical,
title = {{On critical dimensions for compactness in the boundary Yamabe problem, I}},
author = {Liuwei Gong and Seunghyeok Kim and Monica Musso and Juncheng Wei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39945},
eprint = {2609.39945},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39945},
}
RIS
TY - UNPB
TI - On critical dimensions for compactness in the boundary Yamabe problem, I
AU - Liuwei Gong
AU - Seunghyeok Kim
AU - Monica Musso
AU - Juncheng Wei
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39945
UR - https://arxiv.org/abs/2609.39945
AB - We construct smooth, non-locally-conformally-flat metrics on the closed ball for which the boundary Yamabe equation admits L^∞-unbounded sequences of positive solutions. The background metric and the prescribed scalar and boundary mean curvatures are fixed along each sequence. For zero scalar curvature and positive boundary mean curvature, such examples exist with umbilic boundary in every dimension N≥22 and with nonumbilic boundary in every N≥15. For positive scalar curvature and minimal boundary, the corresponding ranges are N≥21 and N≥15. For every N≥9, examples with either umbilic or nonumbilic boundary also exist when the scalar curvature is N(N−1) and the prescribed boundary mean curvature is below a negative threshold depending on N. The construction uses polynomial metric perturbations and corrected bubbles on the half-space, followed by conformal compactification to the ball. We evaluate or estimate the correction solving the linearized Neumann or Robin boundary problem and its contribution to the quadratic term of the reduced energy. We prove that this quadratic term has negative, nondegenerate local extrema with respect to tangential translations and scale in every stated dimension.
ER -