Envelopes of the Thurston metric on Teichmüller space

For the Thurston (asymmetric) metric on Teichmüller space, the defect from being uniquely geodesic is described by the envelope, defined as the union of geodesics from the initial point to the terminal point. Using the harmonic stretch lines we defined recently, we describe the shape of envelopes as a cone over a cone over a space, defined from a topological invariant of the initial and terminal points. In addition, we show that the envelope is always contractible. We prove that envelopes vary continuously with their endpoints. We also provide a parametrization of out-envelopes and in-envelopes in terms of straightened measured laminations complementary to the prescribed maximally stretched laminations. We extend most of these results to the metrically infinite envelopes which have a terminal point on the Thurston boundary, illustrating some of the nuances of these with examples, and describing the accumulation set. Finally, we develop a new characterization of harmonic stretch lines that avoids a limiting process.