Abstract: 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 and describe some generic behavior. We then talk about the connection between the Thurston norm, best Lipschitz circle-valued maps, and maximal stretch laminations, building on the recent work of Daskalopoulos and Uhlenbeck, and Farre, Landesberg, and Minsky. 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.
