The $L^p$-diameter of the space of contractible loops
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910191225667584 |
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| author | Brandenbursky, Michael Shelukhin, Egor |
| author_facet | Brandenbursky, Michael Shelukhin, Egor |
| contents | We prove that the space of contractible simple loops of a given fixed area in any compact oriented surface has infinite diameter as a homogeneous space of the group of area-preserving diffeomorphisms endowed with the $L^p$-metric. As a special case, this resolves the $L^p$-metric analogue of the well-known question in symplectic topology regarding the space of equators on the two-sphere. Our methods involve a new class of functionals on a normed group, which are more general than quasi-morphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07270 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $L^p$-diameter of the space of contractible loops Brandenbursky, Michael Shelukhin, Egor Geometric Topology Differential Geometry Symplectic Geometry We prove that the space of contractible simple loops of a given fixed area in any compact oriented surface has infinite diameter as a homogeneous space of the group of area-preserving diffeomorphisms endowed with the $L^p$-metric. As a special case, this resolves the $L^p$-metric analogue of the well-known question in symplectic topology regarding the space of equators on the two-sphere. Our methods involve a new class of functionals on a normed group, which are more general than quasi-morphisms. |
| title | The $L^p$-diameter of the space of contractible loops |
| topic | Geometric Topology Differential Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2509.07270 |