TITLE: Willmore minimizers for genus $\geq 2$
QUESTION [7 upvotes]: For an immersed closed surface $f: \Sigma \rightarrow \mathbb R^3$ the Willmore functional is defined as
$$
\cal W(f) = \int _{\Sigma}  \frac{1}{4}  |\vec H|^2   d \mu_g,
$$
where $\vec H$ is the mean curvature vector in $\mathbb R^3$and $g$ is the induced metric.
If $\Sigma$ is closed we have the estimate
$$
\cal W(f) \geq 4 \pi
$$
with equality only for $f$ parametrizing a round sphere. 
Recently, the Willmore conjecture was proved (the paper can be found on arxiv), which states that for closed surfaces $\Sigma$ of genus $g \geq 1$ this estimate can be improved:
$$
\cal W(f) \geq 2 \pi^2
$$
with equality only for the Cilfford torus. 
Are there any conjectures about the minimizers in the case of genus $g \geq 2$? And what happens if we consider surfaces immersed in some $\mathbb R^n$ instead of $\mathbb R ^3$?

REPLY [3 votes]: I remember there is a paper by Kusner named: comparison surfaces for the Willmore problem in which the author 
conjectured that the Lawson surface(see Sebastian's answer) minimizes the Willmore energy of genus g surface. For surfaces immersed in R^n, it is also conjectured the Clifford torus should be the minimizer, but it seems to me that this is still an open question.