PreprintarXiv licencearXiv
Uniqueness of self-shrinking ends asymptotic to a cylindrical cone
Abstract
Let Cm ⊂ ℝm+n be a regular embedded cone, where n≥ 1, and let k≥ 1. We prove that two smooth, connected, properly embedded self-shrinking ends whose blow-downs converge locally smoothly to the same cylindrical cone C×ℝk away from the axis {0}×ℝk must coincide. As a consequence, we obtain a rigidity result for complete self-shrinkers asymptotic to cylindrical cones. The proof uses Gaussian decay estimates and a sectorial Carleman estimate to establish backward uniqueness.
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BibTeX
@misc{liu2026uniqueness,
title = {{Uniqueness of self-shrinking ends asymptotic to a cylindrical cone}},
author = {Yile Liu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02303},
eprint = {2611.02303},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02303},
}
RIS
TY - UNPB
TI - Uniqueness of self-shrinking ends asymptotic to a cylindrical cone
AU - Yile Liu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02303
UR - https://arxiv.org/abs/2611.02303
AB - Let C^m ⊂ ℝ^m+n be a regular embedded cone, where n≥ 1, and let k≥ 1. We prove that two smooth, connected, properly embedded self-shrinking ends whose blow-downs converge locally smoothly to the same cylindrical cone C×ℝ^k away from the axis {0}×ℝ^k must coincide. As a consequence, we obtain a rigidity result for complete self-shrinkers asymptotic to cylindrical cones. The proof uses Gaussian decay estimates and a sectorial Carleman estimate to establish backward uniqueness.
ER -