First explicit constrained Willmore minimizers of non-rectangular conformal class

We study immersed tori in $3$-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes $(0,b)$ with $b\sim 1$ the homogenous tori $f^b$ are known to be the unique constrained Willmore minimizers (up to invariance). In this paper we generalize this result and show that the candidates constructed in \cite{HelNdi2} are indeed constrained Willmore minimizers in certain non-rectangular conformal classes $(a,b)\mathrm{.}$ Difficulties arise from the fact that these minimizers are non-degenerate for $a\mathrm{\ne }0$ but smoothly converge to the degenerate homogenous tori $f^b$ as $;a \longrightarrow 0.;$Asabyproductofourarguments,weshowthattheminimalWillmoreenergy$;\omega(a,b);$isrealanalyticandconcavein$;a \in (0, a^b);$forsome$;a^b>0;$andfixed$;b \sim 1,;$ $b\mathrm{\ne }1.$