On the cut locus of Hamilton--Jacobi equations I: structure and propagation via the touching approach
Abstract
For a semiconcave function with linear modulus, we introduce the cut locus through a touching approach and prove that it coincides with the variational cut locus defined via the Lax--Oleinik semigroup, independently of the Hamiltonian. Cut points are characterized by the emptiness of the proximal subdifferential of ϕ, which we use as the analytic criterion behind the touching definition. We introduce the degree of regularity Rϕ, which measures the C1,1 deviation of ϕ, and prove the quantitative relation Rϕ(x)∼ 1/τϕ,H(x) with the cut time function, valid pointwise up to explicit truncation constants. From this relation we derive a separation estimate for calibrated curves, showing that no conjugate points occur before the cut locus. This estimate supplies the tools for the propagation results. Cut points propagate globally along generalized characteristics for the evolutionary Hamilton--Jacobi equation, and Alexandrov points propagate forward along calibrated curves, with the second derivative satisfying a matrix Riccati equation. We also show that the cut locus is a Lebesgue null set and give a streamlined proof of Alexandrov's theorem. Finally, we give necessary and sufficient conditions for the cut locus of a weak KAM solution to be closed, in terms of the cut time function, the C1,1 support, and the degree of regularity.
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BibTeX
@misc{cannarsa2026locus,
title = {{On the cut locus of Hamilton--Jacobi equations I: structure and propagation via the touching approach}},
author = {Piermarco Cannarsa and Wei Cheng and Jiahui Hong \and Wenxue Wei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02879},
eprint = {2609.02879},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02879},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the cut locus of Hamilton--Jacobi equations I: structure and propagation via the touching approach
AU - Piermarco Cannarsa
AU - Wei Cheng
AU - Jiahui Hong \and Wenxue Wei
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02879
UR - https://arxiv.org/abs/2609.02879
AB - For a semiconcave function with linear modulus, we introduce the cut locus through a touching approach and prove that it coincides with the variational cut locus defined via the Lax--Oleinik semigroup, independently of the Hamiltonian. Cut points are characterized by the emptiness of the proximal subdifferential of ϕ, which we use as the analytic criterion behind the touching definition. We introduce the degree of regularity R_ϕ, which measures the C^1,1 deviation of ϕ, and prove the quantitative relation R_ϕ(x)∼ 1/τ_ϕ,H(x) with the cut time function, valid pointwise up to explicit truncation constants. From this relation we derive a separation estimate for calibrated curves, showing that no conjugate points occur before the cut locus. This estimate supplies the tools for the propagation results. Cut points propagate globally along generalized characteristics for the evolutionary Hamilton--Jacobi equation, and Alexandrov points propagate forward along calibrated curves, with the second derivative satisfying a matrix Riccati equation. We also show that the cut locus is a Lebesgue null set and give a streamlined proof of Alexandrov's theorem. Finally, we give necessary and sufficient conditions for the cut locus of a weak KAM solution to be closed, in terms of the cut time function, the C^1,1 support, and the degree of regularity.
ER -