Hausdorff Stability of the Cut Locus Under $C^2$-Perturbations of the Metric
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| Format: | Preprint |
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2025
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| _version_ | 1866912754639568896 |
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| author | Bhowmick, Aritra Itoh, Jin-ichi Prasad, Sachchidanand |
| author_facet | Bhowmick, Aritra Itoh, Jin-ichi Prasad, Sachchidanand |
| contents | In this article, we prove the stability with respect to the Hausdorff metric $d_H$ of the cut locus $\mathrm{Cut}(p, \mathfrak{g})$ of a point $p$ in a compact Riemannian manifold $(M, \mathfrak{g})$ under $C^2$ perturbation of the metric. Specifically, given a sequence of metrics $\mathfrak{g}_i$ on $M$, converging to $\mathfrak{g}$ in the $C^2$ topology, and a sequence of points $p_i$ in $M$, converging to $p$, we show that $\lim_i d_{H}\left( \mathrm{Cut}(p_i, \mathfrak{g}_i), \mathrm{Cut}(p, \mathfrak{g}) \right) = 0$. Along the way, we also prove the continuous dependence of the cut time map on the metric. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_19413 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hausdorff Stability of the Cut Locus Under $C^2$-Perturbations of the Metric Bhowmick, Aritra Itoh, Jin-ichi Prasad, Sachchidanand Differential Geometry Primary: 53C22, 35D40, Secondary: 35F21, 49J52 In this article, we prove the stability with respect to the Hausdorff metric $d_H$ of the cut locus $\mathrm{Cut}(p, \mathfrak{g})$ of a point $p$ in a compact Riemannian manifold $(M, \mathfrak{g})$ under $C^2$ perturbation of the metric. Specifically, given a sequence of metrics $\mathfrak{g}_i$ on $M$, converging to $\mathfrak{g}$ in the $C^2$ topology, and a sequence of points $p_i$ in $M$, converging to $p$, we show that $\lim_i d_{H}\left( \mathrm{Cut}(p_i, \mathfrak{g}_i), \mathrm{Cut}(p, \mathfrak{g}) \right) = 0$. Along the way, we also prove the continuous dependence of the cut time map on the metric. |
| title | Hausdorff Stability of the Cut Locus Under $C^2$-Perturbations of the Metric |
| topic | Differential Geometry Primary: 53C22, 35D40, Secondary: 35F21, 49J52 |
| url | https://arxiv.org/abs/2503.19413 |