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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2208.14517 |
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| _version_ | 1866912377027428352 |
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| author | Kangasniemi, Ilmari Prywes, Eden |
| author_facet | Kangasniemi, Ilmari Prywes, Eden |
| contents | We define a type of modulus $\operatorname{dMod}_p$ for Lipschitz surfaces based on $L^p$-integrable measurable differential forms, generalizing the vector modulus of Aikawa and Ohtsuka. We show that this modulus satisfies a homological duality theorem, where for Hölder conjugate exponents $p, q \in (1, \infty)$, every relative Lipschitz $k$-homology class $c$ has a unique dual Lipschitz $(n-k)$-homology class $c'$ such that $\operatorname{dMod}_p^{1/p}(c) \operatorname{dMod}_q^{1/q}(c') = 1$ and the Poincaré dual of $c$ maps $c'$ to 1. As $\operatorname{dMod}_p$ is larger than the classical surface modulus $\operatorname{Mod}_p$, we immediately recover a more general version of the estimate $\operatorname{Mod}_p^{1/p}(c) \operatorname{Mod}_q^{1/q}(c') \leq 1$, which appears in works by Freedman and He and by Lohvansuu. Our theory is formulated in the general setting of Lipschitz Riemannian manifolds, though our results appear new in the smooth setting as well. We also provide a characterization of closed and exact Sobolev forms on Lipschitz manifolds based on integration over Lipschitz $k$-chains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_14517 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Moduli of Lipschitz Homology Classes Kangasniemi, Ilmari Prywes, Eden Differential Geometry Analysis of PDEs Metric Geometry Primary 31B15, Secondary 28A75, 512F30 We define a type of modulus $\operatorname{dMod}_p$ for Lipschitz surfaces based on $L^p$-integrable measurable differential forms, generalizing the vector modulus of Aikawa and Ohtsuka. We show that this modulus satisfies a homological duality theorem, where for Hölder conjugate exponents $p, q \in (1, \infty)$, every relative Lipschitz $k$-homology class $c$ has a unique dual Lipschitz $(n-k)$-homology class $c'$ such that $\operatorname{dMod}_p^{1/p}(c) \operatorname{dMod}_q^{1/q}(c') = 1$ and the Poincaré dual of $c$ maps $c'$ to 1. As $\operatorname{dMod}_p$ is larger than the classical surface modulus $\operatorname{Mod}_p$, we immediately recover a more general version of the estimate $\operatorname{Mod}_p^{1/p}(c) \operatorname{Mod}_q^{1/q}(c') \leq 1$, which appears in works by Freedman and He and by Lohvansuu. Our theory is formulated in the general setting of Lipschitz Riemannian manifolds, though our results appear new in the smooth setting as well. We also provide a characterization of closed and exact Sobolev forms on Lipschitz manifolds based on integration over Lipschitz $k$-chains. |
| title | On the Moduli of Lipschitz Homology Classes |
| topic | Differential Geometry Analysis of PDEs Metric Geometry Primary 31B15, Secondary 28A75, 512F30 |
| url | https://arxiv.org/abs/2208.14517 |