By Mostow rigidity, every closed hyperbolic 3-manifold supports a unique hyperbolic structure (up to isometry). So all invariants of the manifold defined by the hyperbolic metric are topological invariants. In particular, norms of harmonic representatives of (de Rham) cohomology classes are topological invariants.
In this paper, the authors compare two norms on the first cohomology of closed hyperbolic 3-manifolds: the Thurston norm defined by topology and the \(L^2\) harmonic norm defined by geometry. In particular, the authors prove that, for any closed hyperbolic 3-manifold \(M\) and any \(\phi\in H^1(M;\mathbb{R})\), the following inequality holds: \[ \frac{\pi}{\sqrt{\text{vol}(M)}}\|\phi\|_{Th}\leq \|\phi\|_{L^2}\leq \frac{10\pi}{\sqrt{\text{inj}(M)}}\|\phi\|_{Th}. \] Here \(\text{vol}(M)\) is the hyperbolic volume of \(M\) and \(\text{inj}(M)\) is the injectivity radius of \(M\).
The proof of \(\|\phi\|_{L^2}\geq \frac{\pi}{\sqrt{\text{vol}(M)}}\|\phi\|_{Th}\) uses two auxiliary norms: the least area norm and the harmonic \(L^1\)-norm. The least area norm is closely related with the Thurston norm, while the harmonic \(L^1\)-norm is closely related with the harmonic \(L^2\)-norm. The authors prove the desired inequality by proving these two auxiliary norms are equal to each other.
For the other inequality \(\|\phi\|_{L^2}\leq \frac{10\pi}{\sqrt{\text{inj}(M)}}\|\phi\|_{Th}\), the authors need a control of \(L^{\infty}\)-norms of harmonic forms by their \(L^2\)-norms. They first get such a control on balls in \(\mathbb{H}^3\): for any harmonic 1-form \(\alpha\) in a ball \(B=B(r)\) with radius \(r\) and center \(p\), \[ |\alpha_p|\leq \frac{1}{\sqrt{\nu(r)}}\|\alpha\|_{L^2(B)} \] holds, where \(\nu(r)\) is an explicit function of \(r\). By using a deep result of M. Culler and P. B. Shalen [Isr. J. Math. 190, 445–475 (2012; Zbl 1257.57022)] that 0.29 is a Margulis number for all hyperbolic \(3\)-manifolds with positive \(b_1\), and applying the above estimation on balls, the authors derive that \[ \|\alpha\|_{L^{\infty}}\leq \frac{5}{\sqrt{\text{inj}(M)}}\|\alpha\|_{L^2} \] for any harmonic 1-form on a closed hyperbolic 3-manifold \(M\). Then this inequality implies the desired inequality by a simple calculation.
The authors also give examples to show that some quantitative aspects of the above inequalities are sharp. For the inequality \(\|\phi\|_{L^2}\geq \frac{\pi}{\sqrt{\text{vol}(M)}}\|\phi\|_{Th}\), only the constant \(\pi\) can be improved; for the inequality \(\|\phi\|_{L^2}\leq \frac{10\pi}{\sqrt{\text{inj}(M)}}\|\phi\|_{Th}\), there are examples that \(\frac{\|\phi_n\|_{L^2}}{\|\phi_n\|_{Th}}\to \infty\) like \(\sqrt{-\log{(\text{inj}(M_n))}}\) as \(\text{inj}(M_n)\to 0\).
Another interesting family of closed hyperbolic 3-manifolds is also constructed in this paper. The authors construct a family of hyperbolic 3-manifolds \(M_n\) with universal lower bound on the injectivity radii, with \(b_1(M_n)=1\), and the Thurston norm of the generators of \(H^1(M_n;\mathbb{Z})\) has exponential growth with respect to the volume of manifold. Since the routine estimate implies that there is always an integer cohomology class whose Thurston norm is bounded by an exponential function of the volume, this family of examples shows that this estimate can not be substantially improved. This family of examples is also related with the famous conjecture that the asymptotic growth of homological torsion of congruence covers of a fixed arithmetic hyperbolic 3-manifold equals its hyperbolic volume divided by \(6\pi\).