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