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

TL;DR

This paper establishes bounds on the $L^2$-norm of harmonic 1-forms in hyperbolic 3-manifolds using topological complexity, extending previous inequalities and exploring the interaction between minimal surfaces and harmonic forms.

Contribution

It generalizes inequalities relating harmonic forms and topology in hyperbolic 3-manifolds and introduces new functionals to unify various results and pose open questions.

Findings

Bound on $L^2$-norm of harmonic 1-forms by Thurston norm

Analysis of inequality sharpness in closed and cusped cases

Introduction of functionals unifying results and guiding conjectures

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 $M$. It generalizes two inequalities of Brock-Dunfield. We also study the sharpness of the inequalities in the closed and cusped cases, using the interaction of minimal surfaces and harmonic forms. We unify various results by defining two functionals on orientable closed and cusped hyperbolic $3$-manifolds, and formulate several questions and conjectures.