Monotonicity of infimum of the Willmore energy with prescribed genus

Question

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 $\geq 2$ for a related question.

Answer 1

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, Q. J. Math., Oxf. II. Ser. 37, 437-447 (1986). ZBL0627.53044.

I found 19 papers citing this paper on Google scholar. Of these, one is a survey paper by Kuwert and Schätzle 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.

Answer 2

Of course, the following does not answer your question, but in few of the Kusner conjecture that the Lawson surface $\xi_{g,1}$ should minimize the Willmore energy for closed surface of genus $g$, it still might be interesting: it can be proven that the Willmore energy $\mathcal W(g)$ of $\xi_{g,1}$ has the asymptotic behaviour $$\mathcal W(g)=8\pi\, (1-\frac{\ln{2}}{2g}+\frac{\ln{2}}{2g^2}+O(\frac{1}{g^3})\, ).$$ In particular, the Willmore energy of the Lawson surfaces $\xi_{g,1}$ is increasing for $g\geq N_0\in\mathbb N.$