Epstein Surfaces, $W$-Volume, and the Osgood-Stowe Differential
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911930796474368 |
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| author | Bridgeman, Martin Bromberg, Kenneth |
| author_facet | Bridgeman, Martin Bromberg, Kenneth |
| contents | In a seminal paper, Epstein introduced the theory of what are now called Epstein surfaces, which construct surfaces in $\mathbb{H}^3$ associated to a conformal metric on a domain in $\hat{\mathbb{C}}$. More recently, these surfaces have been used by Krasnov-Schlenker to define the W-volume and renormalized volume associated with a convex co-compact hyperbolic 3-manifold. In this paper we consider Epstein surfaces, W-volume and renormalized volume in two main parts. In the first, we develop an alternate construction of Epstein surfaces using the Osgood-Stowe differential, a generalization of the Schwarzian derivative. Krasnov-Schlenker showed that the metric and shape operator of a surface in hyperbolic space is naturally dual to a conformal metric and shape operator on a projective structure via the hyperbolic Gauss map. We show that this projective shape operator can be derived from the Osgood-Stowe differential. This approach allows us to give a comprehensive and self-contained development of Epstein surfaces, W-volume, and renormalized volume. In the second part, we use the theory developed in the first to prove a number of new results including a generalization of Epstein's univalence criterion, a variational formula for W-volume in terms of the Osgood-Stowe differential, and a use of W-volume to relate the length of the bending lamination of the convex core to the norm of the Schwarzian derivative of the associated univalent map on the conformal boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15065 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Epstein Surfaces, $W$-Volume, and the Osgood-Stowe Differential Bridgeman, Martin Bromberg, Kenneth Differential Geometry Geometric Topology 32G15, 32Q45, 51P05 In a seminal paper, Epstein introduced the theory of what are now called Epstein surfaces, which construct surfaces in $\mathbb{H}^3$ associated to a conformal metric on a domain in $\hat{\mathbb{C}}$. More recently, these surfaces have been used by Krasnov-Schlenker to define the W-volume and renormalized volume associated with a convex co-compact hyperbolic 3-manifold. In this paper we consider Epstein surfaces, W-volume and renormalized volume in two main parts. In the first, we develop an alternate construction of Epstein surfaces using the Osgood-Stowe differential, a generalization of the Schwarzian derivative. Krasnov-Schlenker showed that the metric and shape operator of a surface in hyperbolic space is naturally dual to a conformal metric and shape operator on a projective structure via the hyperbolic Gauss map. We show that this projective shape operator can be derived from the Osgood-Stowe differential. This approach allows us to give a comprehensive and self-contained development of Epstein surfaces, W-volume, and renormalized volume. In the second part, we use the theory developed in the first to prove a number of new results including a generalization of Epstein's univalence criterion, a variational formula for W-volume in terms of the Osgood-Stowe differential, and a use of W-volume to relate the length of the bending lamination of the convex core to the norm of the Schwarzian derivative of the associated univalent map on the conformal boundary. |
| title | Epstein Surfaces, $W$-Volume, and the Osgood-Stowe Differential |
| topic | Differential Geometry Geometric Topology 32G15, 32Q45, 51P05 |
| url | https://arxiv.org/abs/2402.15065 |