  We study the relationship between two norms on the first cohomology of a
hyperbolic 3-manifold: the purely topological Thurston norm and the more
geometric harmonic norm. Refining recent results of Bergeron, \c{S}eng\"un, and
Venkatesh as well as older work of Kronheimer and Mrowka, we show that these
norms are roughly proportional with explicit constants depending only on the
volume and injectivity radius of the hyperbolic 3-manifold itself. Moreover, we
give families of examples showing that some (but not all) qualitative aspects
of our estimates are sharp. Finally, we exhibit closed hyperbolic 3-manifolds
where the Thurston norm grows exponentially in terms of the volume and yet
there is a uniform lower bound on the injectivity radius.
