We consider conformal immersions \(f\) from a 2-dimensional manifold \(M\) into the 3-dimensional sphere which are minimal or have constant mean curvature \(H\). These surfaces admit a generalized Weierstrass representation and their Willmore energies, defined as \(\mathcal{W}(f) = \int_{M}(H^2 + 1)dA\), are given in terms of their Weierstrass data.
Following a generalized Whitham flow, this method allows for the construction of high genus examples of the Lawson surfaces \(\xi_{g,1}\) and leads to the following estimate on their areas:
$$\mathcal{A}(\xi_{g,1}) = \mathcal{W}(\xi_{g,1}) = 8\pi\left( 1-\frac{\ln 2}{2(g+1)} + O\left(\frac{1}{(g+1)^3}\right) \right).$$

The project aims at finding a pathway towards an implicit function theorem argument in order to confirm the Kusner conjecture in the case of symmetric compact surfaces of fixed high genus, which states that the minimizer of the Willmore energy among them is the Lawson surface \(\xi_{g,1}\). This will rely on the construction of high genus limits of various families of minimal and constant mean curvature surfaces, together with the computation of their Willmore energies.