Mollifier smoothings of strongly convex C0-Finsler structures
Abstract
Let M be a smooth manifold and TM its tangent bundle. A C0-Finsler structure of M is a continuous function F: TM → [0, ∞) such that restricted to each tangent space is an asymmetric norm. A mollifier smoothing of F is a family of Finsler structures Fε: TM → [0, ∞), parameterized by ε > 0 and constructed using the standard mollifier, such that Fε → F uniformly on compact subsets as ε → 0. In this work, we construct two mollifier smoothings of F. In the first, we assume that F: TM → [0, ∞) is strongly convex, that is, that F restricted to each tangent space is strongly convex. In the second, we assume that M=G is a Lie group and that F:TG → [0, ∞) is strongly convex and left-invariant, and in this case, Fε is also left-invariant. In both cases, we prove that if F is a Finsler structure, objects such as the fundamental tensor, the Chern, Cartan, Hashiguchi and Berwald connections, and the flag curvature of (M, Fε) converge uniformly on compact subsets to the corresponding objects of $(M, F)$ as ε → 0.
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BibTeX
@misc{fukuoka2026mollifier,
title = {{Mollifier smoothings of strongly convex $C^0$-Finsler structures}},
author = {Ryuichi Fukuoka and Anderson Macedo Setti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12108},
eprint = {2610.12108},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12108},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mollifier smoothings of strongly convex C^0-Finsler structures
AU - Ryuichi Fukuoka
AU - Anderson Macedo Setti
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12108
UR - https://arxiv.org/abs/2610.12108
AB - Let M be a smooth manifold and TM its tangent bundle. A C^0-Finsler structure of M is a continuous function F: TM → [0, ∞) such that restricted to each tangent space is an asymmetric norm. A mollifier smoothing of F is a family of Finsler structures F_ε: TM → [0, ∞), parameterized by ε > 0 and constructed using the standard mollifier, such that F_ε → F uniformly on compact subsets as ε → 0. In this work, we construct two mollifier smoothings of F. In the first, we assume that F: TM → [0, ∞) is strongly convex, that is, that F restricted to each tangent space is strongly convex. In the second, we assume that M=G is a Lie group and that F:TG → [0, ∞) is strongly convex and left-invariant, and in this case, F_ε is also left-invariant. In both cases, we prove that if F is a Finsler structure, objects such as the fundamental tensor, the Chern, Cartan, Hashiguchi and Berwald connections, and the flag curvature of (M, F_ε) converge uniformly on compact subsets to the corresponding objects of $(M, F)$ as ε → 0.
ER -