Willmore minimizers for genus $\geq 2$

Question

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 $\mathbb R ^3$?

Answer 1

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 $\xi_{g,1}.$

Answer 2

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.