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 $\mathbb R ^3$?



Willmore minimizers for genus $\geq 2$
First of all, by a result of Bauer and Kuwert, there exists a smooth minimizer of the Willmore functional in the class of compact surfaces with fixed genus g, for any g. They have Willmore functional below $8\pi$ and by a result of Kuwert, Li and Schaetzle, the Willmore functional of the minimzers for genus $g$ tends to $8\pi$ when $g$ goes to infinity.
Not much more is known about higher genus surfaces, but there is a vague conjecture, that the minimzers are the so called Lawson surface $\xi\_{g,1}.$


