On unions of geodesics and projections of invariant sets
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917203344883712 |
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| author | Li, Longhui |
| author_facet | Li, Longhui |
| contents | Let $M$ be a $d$-dimensional complete Riemannian manifold and let $π: SM \to M$ denote the canonical projection from the unit tangent bundle. We prove that if $E \subset SM$ is a set that invariant under the geodesic flow with Hausdorff dimension $\dim_{\mathcal{H}} E \ge 2(k-1)+1 +β$ for some integer $1 \le k \le d-1$ and some $β\in [0,1]$, then the projection $π(E)$ satisfies $\dim_{\mathcal{H}} π(E) \ge k + β$. In other words, this yields a lower bound on the Hausdorff dimension of unions of geodesics in $M$. Our theorem extends a result of J. Zahl concerning unions of lines in $\mathbb{R}^d$. The proof relies on the transversal property of geodesics, an appropriate $(k+1)$-linear curved Kakeya estimate, and the Bourgain-Guth argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09202 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On unions of geodesics and projections of invariant sets Li, Longhui Classical Analysis and ODEs Differential Geometry Let $M$ be a $d$-dimensional complete Riemannian manifold and let $π: SM \to M$ denote the canonical projection from the unit tangent bundle. We prove that if $E \subset SM$ is a set that invariant under the geodesic flow with Hausdorff dimension $\dim_{\mathcal{H}} E \ge 2(k-1)+1 +β$ for some integer $1 \le k \le d-1$ and some $β\in [0,1]$, then the projection $π(E)$ satisfies $\dim_{\mathcal{H}} π(E) \ge k + β$. In other words, this yields a lower bound on the Hausdorff dimension of unions of geodesics in $M$. Our theorem extends a result of J. Zahl concerning unions of lines in $\mathbb{R}^d$. The proof relies on the transversal property of geodesics, an appropriate $(k+1)$-linear curved Kakeya estimate, and the Bourgain-Guth argument. |
| title | On unions of geodesics and projections of invariant sets |
| topic | Classical Analysis and ODEs Differential Geometry |
| url | https://arxiv.org/abs/2601.09202 |