Mollifier smoothing of left-invariant strongly convex $C^0$-Finsler structures on Lie groups and convergence of extremals
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| Format: | Preprint |
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2025
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| _version_ | 1866911237129895936 |
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| author | Fukuoka, Ryuichi Setti, Anderson Macedo |
| author_facet | Fukuoka, Ryuichi Setti, Anderson Macedo |
| contents | Let $M$ be a smooth manifold and $TM$ its tangent bundle. A $C^0$-Finsler structure of $M$ is a continuous function $F:TM \rightarrow \mathbb{R}$ such that $F$ restricted to each tangent space $T_xM$ of $M$ is an asymmetric norm. $F$ is strongly convex if $F\vert_{T_xM}$ is a strongly convex asymmetric norm for every $x \in M$. Let $G$ be a Lie group endowed with a left-invariant strongly convex $C^0$-Finsler structure $F$.
We introduce a smoothing $F_{\varepsilon}$ of $F$, which is a left-invariant version of the mollifier smoothing presented previously by the same authors.
We study extremals $x(t)$ on $(G,F)$ using the Pontryagin maximum principle.
Given $(x_0,α_0)$ in the cotangent bundle $T^\ast G$ of $G$, we prove that there exist a unique Pontryagin extremal $t\in \mathbb{R} \mapsto (x(t), α(t))$ such that $(x(0),α(0))=(x_0,α_0)$.
Moreover, if $t \in \mathbb{R} \mapsto (x_\varepsilon(t), α_{\varepsilon}(t))$ is the unique Pontryagin extremal on $(G,F_\varepsilon)$ such that $(x_\varepsilon(0), α_{\varepsilon}(0))=(x_0, α_0)$, then we prove that $(x_{\varepsilon}(t),α_\varepsilon(t))$ converges uniformly to $(x(t),α(t))$ on compact intervals of $\mathbb{R}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_24666 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mollifier smoothing of left-invariant strongly convex $C^0$-Finsler structures on Lie groups and convergence of extremals Fukuoka, Ryuichi Setti, Anderson Macedo Differential Geometry 49N99, 53B20, 53B99, 53C22, 53D25 Let $M$ be a smooth manifold and $TM$ its tangent bundle. A $C^0$-Finsler structure of $M$ is a continuous function $F:TM \rightarrow \mathbb{R}$ such that $F$ restricted to each tangent space $T_xM$ of $M$ is an asymmetric norm. $F$ is strongly convex if $F\vert_{T_xM}$ is a strongly convex asymmetric norm for every $x \in M$. Let $G$ be a Lie group endowed with a left-invariant strongly convex $C^0$-Finsler structure $F$. We introduce a smoothing $F_{\varepsilon}$ of $F$, which is a left-invariant version of the mollifier smoothing presented previously by the same authors. We study extremals $x(t)$ on $(G,F)$ using the Pontryagin maximum principle. Given $(x_0,α_0)$ in the cotangent bundle $T^\ast G$ of $G$, we prove that there exist a unique Pontryagin extremal $t\in \mathbb{R} \mapsto (x(t), α(t))$ such that $(x(0),α(0))=(x_0,α_0)$. Moreover, if $t \in \mathbb{R} \mapsto (x_\varepsilon(t), α_{\varepsilon}(t))$ is the unique Pontryagin extremal on $(G,F_\varepsilon)$ such that $(x_\varepsilon(0), α_{\varepsilon}(0))=(x_0, α_0)$, then we prove that $(x_{\varepsilon}(t),α_\varepsilon(t))$ converges uniformly to $(x(t),α(t))$ on compact intervals of $\mathbb{R}$. |
| title | Mollifier smoothing of left-invariant strongly convex $C^0$-Finsler structures on Lie groups and convergence of extremals |
| topic | Differential Geometry 49N99, 53B20, 53B99, 53C22, 53D25 |
| url | https://arxiv.org/abs/2510.24666 |