Quantitative avoidance for free boundary flows and applications
Abstract
In this article, we introduce a new distance function between hypersurfaces with free boundary. We show that our new quantity, which we call twisted Fermi distance, is monotone under mean curvature flow with free boundary. This overcomes the stumbling block that monotonicity of the usual distance function can fail for non-convex domains, and has several applications. Most importantly, we generalize the avoidance principle for free boundary Brakke flows, recently established by the first author for convex domains, to arbitrary domains. Using this, we then show that all results from our recent joint work, including the mean-convex neighborhood theorem and the uniqueness theorem for free boundary flows through cylindrical singularities, can be generalized to arbitrary domains without any convexity assumptions as well.
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BibTeX
@misc{bao2026quantitative,
title = {{Quantitative avoidance for free boundary flows and applications}},
author = {Yueheng Bao and Robert Haslhofer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01026},
eprint = {2609.01026},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01026},
}
RIS
TY - UNPB
TI - Quantitative avoidance for free boundary flows and applications
AU - Yueheng Bao
AU - Robert Haslhofer
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01026
UR - https://arxiv.org/abs/2609.01026
AB - In this article, we introduce a new distance function between hypersurfaces with free boundary. We show that our new quantity, which we call twisted Fermi distance, is monotone under mean curvature flow with free boundary. This overcomes the stumbling block that monotonicity of the usual distance function can fail for non-convex domains, and has several applications. Most importantly, we generalize the avoidance principle for free boundary Brakke flows, recently established by the first author for convex domains, to arbitrary domains. Using this, we then show that all results from our recent joint work, including the mean-convex neighborhood theorem and the uniqueness theorem for free boundary flows through cylindrical singularities, can be generalized to arbitrary domains without any convexity assumptions as well.
ER -