For closed hyperbolic 3-manifolds, Brock and Dunfield prove an inequality on the first cohomology bounding the ratio of the geometric L2-norm to the topological Thurston norm. Motivated by Dehn fillings, they conjecture that as the injectivity radius tends to 0, the ratio is big O of the square root of the log of the injectivity radius. We prove this conjecture for all sequences of manifolds which geometrically converge. Generically, we prove that the ratio is bounded by a constant, by showing that any least area of a closed surface is disjoint from the thin part. We then study 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.
Note: The seminar is online this week.