Xiaolong Hans Han

For closed hyperbolic 3-manifolds, Brock and Dunfield made a conjecture about the upper bound on the ratio of L2-norm to Thurston norm. We first talk about its proof assuming manifolds have bounded volume and describe some generic behavior. We then talk about the connection between the Thurston norm, best Lipschitz circle-valued maps, and maximal stretch laminations. We show that the distance between a level set and its translation is the reciprocal of the Lipschitz constant, bounded by the topological entropy of the pseudo-Anosov monodromy if M fibers.