The Weil-Peterson Hessian of Length on Teichmuller Space
| dc.citation.firstpage | 129 | en_US |
| dc.citation.journalTitle | Journal of Differential Geometry | en_US |
| dc.citation.lastpage | 169 | en_US |
| dc.citation.volumeNumber | 91 | en_US |
| dc.contributor.author | Wolf, Michael | en_US |
| dc.date.accessioned | 2013-09-13T15:16:10Z | |
| dc.date.available | 2013-09-13T15:16:10Z | |
| dc.date.issued | 2012 | en_US |
| dc.description.abstract | We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this functional over Teichmuller space (due to Wolpert (1987)). We then estimate this Hessian from below in terms of local quantities and distance along the geodesic. The formula extends to proper arcs on punctured hyperbolic surfaces, and the estimate to laminations. Wolpert’s result that the Thurston metric is a multiple of the Weil-Petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves. We give further applications to upper bounds of the Hessian, especially near pinching loci, recover through a geometric argumentWolpert’s result on the convexity of length to the half-power, and give a lower bound for growth of length in terms of twist. | en_US |
| dc.embargo.terms | none | en_US |
| dc.identifier.citation | Wolf, Michael. "The Weil-Peterson Hessian of Length on Teichmuller Space." <i>Journal of Differential Geometry,</i> 91, (2012) International Press: 129-169. <a href="https://hdl.handle.net/1911/71888">https://hdl.handle.net/1911/71888</a>. | |
| dc.identifier.uri | https://hdl.handle.net/1911/71888 | |
| dc.language.iso | eng | en_US |
| dc.publisher | International Press | |
| dc.rights | Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. | |
| dc.title | The Weil-Peterson Hessian of Length on Teichmuller Space | en_US |
| dc.type | Journal article | en_US |
| dc.type.dcmi | Text | en_US |
| dc.type.publication | publisher version | en_US |