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Unbounded total mean curvature with prescribed boundary metric
Abstract
Bär has shown that a fixed boundary metric and a scalar-curvature lower bound do not generally bound total mean curvature from above. We extend this unboundedness to every smooth compact connected manifold of dimension n≥3 with nonempty boundary and every prescribed boundary metric, using scalar-flat metrics. While Bär's examples use warped products, our construction uses a nonzero cubic variation of total mean curvature near a flat product, followed by localization and conformal correction. We also construct arbitrarily Ck-small perturbations of the flat torical half-cylinder with nonnegative scalar curvature, unchanged boundary metric, an exact product end, and positive total boundary mean curvature.
Related
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BibTeX
@misc{wang2026unbounded,
title = {{Unbounded total mean curvature with prescribed boundary metric}},
author = {Zhixin Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04762},
eprint = {2611.04762},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04762},
}
RIS
TY - UNPB
TI - Unbounded total mean curvature with prescribed boundary metric
AU - Zhixin Wang
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04762
UR - https://arxiv.org/abs/2611.04762
AB - Bär has shown that a fixed boundary metric and a scalar-curvature lower bound do not generally bound total mean curvature from above. We extend this unboundedness to every smooth compact connected manifold of dimension n≥3 with nonempty boundary and every prescribed boundary metric, using scalar-flat metrics. While Bär's examples use warped products, our construction uses a nonzero cubic variation of total mean curvature near a flat product, followed by localization and conformal correction. We also construct arbitrarily C^k-small perturbations of the flat torical half-cylinder with nonnegative scalar curvature, unchanged boundary metric, an exact product end, and positive total boundary mean curvature.
ER -