We study surface energies depending on the mean curvature in total spaces of Killing submersions, which extend the classical notion of Willmore energy. Based on a symmetry reduction procedure, we construct vertical tori critical for these mean curvature energies. These vertical tori are based on closed curves critical for curvature energy functionals in Riemannian 2-space forms.

The binormal evolution of these critical curves in Riemannian 3-space forms generates rotational tori solutions for an ample family of Weingarten surfaces. Therefore, we also introduce some correspondence results between these two types of tori and illustrate their relation.

我们根据Killing浸没空间的平均曲率研究表面能，这扩展了Willmore能量的经典概念。基于对称减少程序，我们构造了这些平均曲率能量的临界垂直托里。这些垂直托里基于对黎曼2空间形式的曲率能量函数至关重要的闭合曲线。

这些临界曲线以黎曼3空间形式的双正态演化产生了足够的Weingarten曲面族的旋转托里解。因此，我们还介绍了这两种类型的花托之间的对应关系，并说明了它们之间的关系。