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

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.

Incorporated referee's comments. Accepted by IMRN

References in corpus (3)

• L^2 harmonics forms on non compact manifolds
• Scalar curvature and harmonic one-forms on three-manifolds with boundary
• On the Betti Numbers of Finite Volume Hyperbolic Manifolds