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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.16737 |
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| _version_ | 1866915943413710848 |
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| author | Fan, Shuo Viklund, Fredrik Wang, Yilin |
| author_facet | Fan, Shuo Viklund, Fredrik Wang, Yilin |
| contents | To any Jordan curve one may associate a circle homeomorphism $φ: \mathbb S^1 \to \mathbb S^1$ via conformal welding. Through this correspondence, the Loewner energy $I^L$, also known as the universal Liouville action, is a Kähler potential for the unique homogeneous Kähler metric on the universal Teichmüller space. Despite this, explicit expressions for $I^L$ in terms of $φ$ alone do not seem to be available in the literature.
In this paper, we obtain such formulas. For this, we introduce an operator ${\bf Λ}_φ$ defined using the Fourier coefficients of the function \[ (z,w) \mapsto \log \left|\frac{φ(z)-φ(w)}{z-w}\right|, \qquad (z,w) \in \mathbb{S}^1 \times \mathbb{S}^1. \] We relate ${\bf Λ}_φ$ to the single-layer potential and composition operator, and prove an analog of the classical Grunsky inequalities for quasisymmetric $φ$. We show moreover that $φ$ is Weil--Petersson if and only if ${\bf Λ}_φ$ is Hilbert--Schmidt, and we express $I^L$ as several related Fredholm determinants as well as a regularized Fredholm determinant. We also treat Schatten classes, and we obtain formulas in terms of Dirichlet integrals involving $\log φ'$ and in terms of the composition operator induced by $φ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16737 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Loewner energy of a welding homeomorphism Fan, Shuo Viklund, Fredrik Wang, Yilin Complex Variables To any Jordan curve one may associate a circle homeomorphism $φ: \mathbb S^1 \to \mathbb S^1$ via conformal welding. Through this correspondence, the Loewner energy $I^L$, also known as the universal Liouville action, is a Kähler potential for the unique homogeneous Kähler metric on the universal Teichmüller space. Despite this, explicit expressions for $I^L$ in terms of $φ$ alone do not seem to be available in the literature. In this paper, we obtain such formulas. For this, we introduce an operator ${\bf Λ}_φ$ defined using the Fourier coefficients of the function \[ (z,w) \mapsto \log \left|\frac{φ(z)-φ(w)}{z-w}\right|, \qquad (z,w) \in \mathbb{S}^1 \times \mathbb{S}^1. \] We relate ${\bf Λ}_φ$ to the single-layer potential and composition operator, and prove an analog of the classical Grunsky inequalities for quasisymmetric $φ$. We show moreover that $φ$ is Weil--Petersson if and only if ${\bf Λ}_φ$ is Hilbert--Schmidt, and we express $I^L$ as several related Fredholm determinants as well as a regularized Fredholm determinant. We also treat Schatten classes, and we obtain formulas in terms of Dirichlet integrals involving $\log φ'$ and in terms of the composition operator induced by $φ$. |
| title | On the Loewner energy of a welding homeomorphism |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2604.16737 |