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 homogeneous 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 [14] are indeed constrained Willmore minimizers in certain non-rectangular conformal classes $(a,b)$. Difficulties arise from the fact that these minimizers are non-degenerate for $a\ne 0$ but smoothly converge to the degenerate homogeneous tori $f^b$ as $a⟶0$. As a byproduct of our arguments, we show that the minimal Willmore energy $\omega (a,b)$ is real analytic and concave in $a\in (0,a^b)$ for some $a^b>0$ and fixed $b\sim 1$, $b\ne 1$.

我们研究浸入式环面在 3 空间中最小化各自保形类中的 Willmore 能量。在矩形保形类中$(0,乙)$ 和 $乙～1$ 同质的托里 $F^{乙}$已知是唯一受约束的 Willmore 最小化器（直到不变性）。在本文中，我们概括了这一结果，并表明 [14] 中构造的候选者确实是某些非矩形保形类中的受约束 Willmore 极小值$(\mathrm{一种},乙)$. 困难来自于这些最小化器是非退化的$\mathrm{一种}\ne 0$ 但平滑地收敛到退化的同质环面 $F^{乙}$ 作为 $\mathrm{一种}⟶0$. 作为我们论证的副产品，我们证明了最小的 Willmore 能量$\omega (\mathrm{一种},乙)$ 是实解析且凹的 $\mathrm{一种}\in (0,{\mathrm{一种}}^{乙})$ 对于一些 ${\mathrm{一种}}^{乙}>0$ 并固定 $乙～1$, $乙\ne 1$.