In this work, we investigate the mid-range behavior of geometry along a grafting ray in Teichm\"{u}ller space. The main technique is to describe the hyperbolic metric $\sigma_{t}$ at a point along the grafting ray in terms of a conformal factor $g_{t}$ times the Thurston (grafted) metric and study solutions to the linearized Liouville equation. We give a formula that describes, at any point on a grafting ray, the change in length of a sum of distinguished curves in terms of the hyperbolic geometry at the point. We then make precise the idea that once the length of the grafting locus is small, local behavior of the geometry for grafting on a general manifold is like that of grafting on a cylinder. Finally, we prove that the sum of lengths of is eventually monotone decreasing along grafting rays.
| Identifer | oai:union.ndltd.org:RICE/oai:scholarship.rice.edu:1911/64672 |
| Date | 06 September 2012 |
| Creators | Laverdiere, Renee |
| Contributors | Wolf, Michael |
| Source Sets | Rice University |
| Language | English |
| Detected Language | English |
| Type | thesis, text |
| Format | application/pdf |
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