Let 
$$
\beta\_g:=\inf\{\frac14\int\_\Sigma H^2 d\mu \hspace{0.2cm} | \hspace{0.2cm} \Sigma\subset \mathbb
R^{3}, \operatorname{genus}(\Sigma)=g \}
$$
be the infimum of the Willmore energy of embedded genus-$g$ surfaces. A standard result by Willmore says that $\beta\_0=4\pi$ and Marquez-Neves showed that $\beta\_1=2\pi^2$ as well as $\beta\_g\geq 2\pi^2$ for $g\geq 1$. By estimating the Willmore energy of the Lawson surfaces, one finds $\beta\_g<8\pi$ for all $g$. On the other hand, by a result of Kuwert-Li-Schaetzle, one has $\beta\_g\to 8\pi$ as $g$ tends to infinity. This suggests that $\beta\_g$ is (perhaps strictly) non-decreasing in $g$. I am wondering if this is true?


See also [Willmore minimizers for genus $\geq 2$](https://mathoverflow.net/questions/92013/willmore-minimizers-for-genus-geq-2) for a related question.



Monotonicity of infimum of the Willmore energy with prescribed genus
As pointed out in the paper you reference, this question was posed on p. 446 of this paper 


*Kühnel, Wolfgang; Pinkall, Ulrich*, [**On total mean curvatures**](http://dx.doi.org/10.1093/qmath/37.4.437), Q. J. Math., Oxf. II. Ser. 37, 437-447 (1986). [ZBL0627.53044](https://zbmath.org/?q=an:0627.53044).


I found 19 papers citing this paper on Google scholar. Of these, one is a [survey paper by Kuwert and Schätzle](https://link.springer.com/content/pdf/10.1007/978-88-7642-427-4_1.pdf) from 2012 which reproves the Kuwert-Li-Schäzle result (and doesn't mention the monotonicity question). A cursory look at the titles and abstracts indicates that this question doesn't seem to have been resolved (although there are several papers by Kuwert-Schäzle which address related questions). Presumably a paper proving monotonicity would cite their paper, so I'm guessing that this is still an open question since 1986. 


