# Monotonicity of infimum of the Willmore energy with prescribed genus

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.

• As far as I know, nobody has been able to calculate the Willmore energy for Lawson surfaces of genus bigger than 1, but it is only possible to estimate their energy. – Sebastian Sep 19 '18 at 8:25
• You're completely right, I just edited my post. – user128470 Sep 20 '18 at 15:49