CR-invariant energy of Legendrian knots in the Heisenberg group
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911695679520768 |
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| author | Matsumoto, Yoshihiko O'Hara, Jun |
| author_facet | Matsumoto, Yoshihiko O'Hara, Jun |
| contents | We introduce an energy functional for Legendrian knots in the 3-dimensional Heisenberg group $\mathcal{H}$, which serves as a sub-Riemannian analog of the Möbius invariant knot energy in Euclidean 3-space introduced by the second author. The energy is obtained by regularizing a divergent integral of the potential of order -2 with respect to the Korányi distance on $\mathcal{H}$; this choice of distance is essential for the energy to be invariant under the action of PU(2,1). We characterize $\mathbb{R}$-circles in $\mathcal{H}$ as the minimizers of the energy, and establish a Heisenberg analog of the Doyle--Schramm cosine formula. We also show that the energy integrand admits an expression in terms of a complex-valued 2-form on the complement of the diagonal in $\mathcal{H}\times\mathcal{H}$, providing a partial analog of the infinitesimal cross ratio interpretation known from the classical setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25713 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | CR-invariant energy of Legendrian knots in the Heisenberg group Matsumoto, Yoshihiko O'Hara, Jun Geometric Topology Complex Variables Differential Geometry 57K10 (Primary) 32V05, 57K33 (Secondary) We introduce an energy functional for Legendrian knots in the 3-dimensional Heisenberg group $\mathcal{H}$, which serves as a sub-Riemannian analog of the Möbius invariant knot energy in Euclidean 3-space introduced by the second author. The energy is obtained by regularizing a divergent integral of the potential of order -2 with respect to the Korányi distance on $\mathcal{H}$; this choice of distance is essential for the energy to be invariant under the action of PU(2,1). We characterize $\mathbb{R}$-circles in $\mathcal{H}$ as the minimizers of the energy, and establish a Heisenberg analog of the Doyle--Schramm cosine formula. We also show that the energy integrand admits an expression in terms of a complex-valued 2-form on the complement of the diagonal in $\mathcal{H}\times\mathcal{H}$, providing a partial analog of the infinitesimal cross ratio interpretation known from the classical setting. |
| title | CR-invariant energy of Legendrian knots in the Heisenberg group |
| topic | Geometric Topology Complex Variables Differential Geometry 57K10 (Primary) 32V05, 57K33 (Secondary) |
| url | https://arxiv.org/abs/2604.25713 |