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We establish a link between the behavior of length functions on Teichm\"ullerspace and the geometry of certain anti de Sitter 3-manifolds. As anapplication, we give new purely anti de Sitter proofs of results ofTeichm\"uller theory such as (strict) convexity of length functions along shearpaths and geometric bounds on their second variation along earthquakes. Alongthe way, we provide shear-bend coordinates for Mess' anti de Sitter3-manifolds.

Length functions in Teichm\"uller and anti de Sitter geometry