Thurston norms, $L^2$-norms, geodesic laminations, and Lipschitz maps

Abstract

For closed hyperbolic $3$-manifolds $M$ with volume less than a constant $V$, we prove an inequality regarding the geometric $L^2$-norm and the topological Thurston norm, which is qualitatively sharp and verifies a conjecture of Brock and Dunfield in this case. Generically, we show that the $L^2$-norm is less than a constant $c(V)$ times the Thurston norm by showing that any least area 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. 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. For infinitely many examples constructed by Rudd, we show the entropy is bounded from below by one-third the length of the circumference.

45 pages, 4 figures. Updates: Theorem 1.1 resolves the conjecture for all hyperbolic $3$-manifolds with volume <V. Lemma 1.2 takes into account all short geodesics. Add Lemma 2.1 bounding the intrinsic diameter of the boundary of tubes by V, A.5, and A.6 which show that the Brock-Dunfield conjecture does not reflect the local order, in contrast to their original inequality