Harmonic Forms, Minimal Surfaces, and Norms on Cohomology of Hyperbolic 3-Manifolds

Abstract

Abstract We bound the $L^2$-norm of an $L^2$ harmonic $1$-form in an orientable cusped hyperbolic $3$-manifold $M$ by its topological complexity, measured by the Thurston norm, up to a constant depending on $M$. This generalizes two inequalities of Brock and Dunfield. We also study the sharpness of the inequalities in the closed and cusped cases, using the interaction between minimal surfaces and harmonic forms. We unify various results by defining two functionals on closed and cusped hyperbolic $3$-manifolds and formulate several questions and conjectures.