PreprintCC BY-NC-ND 4.0arXiv
Rigidity and Non-existence Results of λ-Translating Solitons
Abstract
A λ-translating soliton is a hypersurface in ℝn+1 satisfying H=⟨ T,ν⟩+λ; equivalently, it has constant weighted mean curvature with respect to the log-linear density e⟨ T,X⟩, and is an eternal solution of the mean curvature flow with a constant forcing term. In this paper, we first prove that every complete properly immersed λ-translating soliton with λ>0 and infΣH>λ has at least exponential volume growth, in contrast with the linear growth of ordinary translating solitons. We then prove sharp non-existence results for graphic λ-translating solitons (λgeqslant 0) with bounded gradient, and two rigidity theorems.
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BibTeX
@misc{xiang2026rigidity,
title = {{Rigidity and Non-existence Results of $λ$-Translating Solitons}},
author = {Li Xiang and Sun Jun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18064},
eprint = {2609.18064},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18064},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Rigidity and Non-existence Results of λ-Translating Solitons
AU - Li Xiang
AU - Sun Jun
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18064
UR - https://arxiv.org/abs/2609.18064
AB - A λ-translating soliton is a hypersurface in ℝ^n+1 satisfying H=⟨ T,ν⟩+λ; equivalently, it has constant weighted mean curvature with respect to the log-linear density e^⟨ T,X⟩, and is an eternal solution of the mean curvature flow with a constant forcing term. In this paper, we first prove that every complete properly immersed λ-translating soliton with λ>0 and inf_ΣH>λ has at least exponential volume growth, in contrast with the linear growth of ordinary translating solitons. We then prove sharp non-existence results for graphic λ-translating solitons (λgeqslant 0) with bounded gradient, and two rigidity theorems.
ER -