Let $(M,g)$ be a compact Riemannian three-fold such that $H\_2(M,\mathbb{Z}) = \mathbb{Z}$ and $S$ any surface representing 1. By Hodge theory there exist a harmonic differential one-form $\eta$ dual to the surface $S$, meaning for all closed 2-form $\alpha$ we have : 


\begin{equation}
\int\_M \eta \wedge \alpha = \int\_S \alpha 
\end{equation}


and 
\begin{equation}
\Delta \eta = 0
\end{equation}


where $\Delta$ is the Hodge De Rham Laplacian acting on one-forms.


My question is : does it possible to control the $L^1$ and the $L^2$ norm of $\eta$ with geometric quantities ? 


I was thinking about the following : any such $\eta$ gives an application $M \to \mathbb{S}^1$ by integrating $\eta$ on any path and takes the result modulo $\mathbb{Z}$ (it does not depend of the path by the assumptions on $\eta$ to be an integer class and being closed). Therefore, by the co-area formula, I expect the $L^1$ norm to be related to the area of surfaces representing $S$, the minimal one for instance.



harmonic differential form integer class
The norms in cohomology are isometric to suitable norms in homology which can be expressed using geometric quantities, but this is far from trivial. The question is studied in 


Bangert, Victor; Katz, Mikhail.
Stable systolic inequalities and cohomology products.
Dedicated to the memory of Jürgen K. Moser. 
Comm. Pure Appl. Math. 56 (2003), no. 7, 979–997. 


