For every $b>1$ fixed, we explicitly construct 1-dimensional families of embedded constrained Willmore tori parametrized by their conformal class $(a,b)$ with $a{\sim }_b{0}^{+}$ deforming the homogeneous torus $f^b$ of conformal class $(0,b).$ The variational vector field at $f^b$ is hereby given by a non-trivial zero direction of a penalized Willmore stability operator which we show to coincide with a double point of the corresponding spectral curve. Further, we characterize for $b\sim 1$, $b\ne 1$ and $a{\sim }_b{0}^{+}$ the family obtained by opening the “smallest” double point on the spectral curve which is heuristically the direction with the smallest increase of Willmore energy at $f^b$. Indeed we show in Heller and Ndiaye (2021) that these candidates minimize the Willmore energy in their respective conformal class for $b\sim 1$, $b\ne 1$ and $a{\sim }_b{0}^{+}.$

对于每个 $b>\mathrm{1个}$ 固定后，我们明确构造了由其共形类参数化的嵌入式约束Willmore花托的一维族 $（\mathrm{一个}，b）$ 和 $\mathrm{一个}{〜}_b{0}^{+}$ 变形同质圆环 $F^b$ 适形阶级 $（0，b）。$ 处的变分矢量场 $F^b$因此，由受罚的Willmore稳定性算子的一个非平凡的零方向给出，我们证明它与相应光谱曲线的双点重合。此外，我们表征$b〜\mathrm{1个}$， $b\ne \mathrm{1个}$ 和 $\mathrm{一个}{〜}_b{0}^{+}$ 通过打开光谱曲线上的“最小”双点而获得的族，这是启发式地在Willmore能量增加最小的方向 $F^b$。的确，我们在Heller和Ndiaye（2021）中证明，这些候选物在其各自的保形类中将Willmore的能量最小化，$b〜\mathrm{1个}$， $b\ne \mathrm{1个}$ 和 $\mathrm{一个}{〜}_b{0}^{+}。$