Thurston norms, $L^2$-norms, geodesic laminations, and Lipschitz maps
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911788327501824 |
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| author | Han, Xiaolong Hans |
| author_facet | Han, Xiaolong Hans |
| contents | For closed hyperbolic $3$-manifolds $M$ with volume less than a constant $V$, we prove an inequality regarding the geometric $L^2$-norm and the topological Thurston norm, which is qualitatively sharp and verifies a conjecture of Brock and Dunfield in this case. Generically, we show that the $L^2$-norm is less than a constant $c(V)$ times the Thurston norm by showing that any least area closed surface is disjoint from the thin part.
We then study the connection between the Thurston norm, best Lipschitz circle-valued maps, and maximal stretch laminations, building on the recent work of Daskalopoulos and Uhlenbeck, and Farre, Landesberg and Minsky. We show that the distance between a level set and its translation is the reciprocal of the Lipschitz constant, bounded by the topological entropy of the pseudo-Anosov monodromy if $M$ fibers. For infinitely many examples constructed by Rudd, we show the entropy is bounded from below by one-third the length of the circumference. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_18093 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Thurston norms, $L^2$-norms, geodesic laminations, and Lipschitz maps Han, Xiaolong Hans Geometric Topology Differential Geometry 57M50, 57K32, 53A10 For closed hyperbolic $3$-manifolds $M$ with volume less than a constant $V$, we prove an inequality regarding the geometric $L^2$-norm and the topological Thurston norm, which is qualitatively sharp and verifies a conjecture of Brock and Dunfield in this case. Generically, we show that the $L^2$-norm is less than a constant $c(V)$ times the Thurston norm by showing that any least area closed surface is disjoint from the thin part. We then study the connection between the Thurston norm, best Lipschitz circle-valued maps, and maximal stretch laminations, building on the recent work of Daskalopoulos and Uhlenbeck, and Farre, Landesberg and Minsky. We show that the distance between a level set and its translation is the reciprocal of the Lipschitz constant, bounded by the topological entropy of the pseudo-Anosov monodromy if $M$ fibers. For infinitely many examples constructed by Rudd, we show the entropy is bounded from below by one-third the length of the circumference. |
| title | Thurston norms, $L^2$-norms, geodesic laminations, and Lipschitz maps |
| topic | Geometric Topology Differential Geometry 57M50, 57K32, 53A10 |
| url | https://arxiv.org/abs/2310.18093 |