标题： 关于最小惯性矩的平稳曲面的两个分类结果 显示英文标题

标题： Two classification results for stationary surfaces of the least moment of inertia

摘要： 欧几里得空间中的曲面$\r^3$被称为$\alpha$-平稳曲面，如果它是能量$\int_\Sigma|p|^\alpha$的临界点，其中$\alpha\in\r$。 这些曲面由欧拉-拉格朗日方程$H(p)=\alpha\frac{\langle N(p),p\rangle}{|p|^2}$，$p\in\Sigma$表征，其中$H$和$N$分别是曲面$\Sigma$的平均曲率和法向量。如果$\alpha\not=0$，我们证明了向量平面是唯一满足$\alpha$的直纹曲面。 分类的第二个结果断言，如果 $\alpha\not=-2,-4$，任何由圆组成的 $\alpha$-平稳曲面必须是旋转曲面。 如果 $\alpha=-4$，该曲面是平面、螺旋面、悬链面或黎曼极小例的反演。 如果 $\alpha=-2$，并且除了以 $0$为中心的球面外，我们发现非球形的循环 $-$-平稳曲面。 显示英文摘要

摘要： A surface in Euclidean space $\r^3$ is said to be an $\alpha$-stationary surface if it is a critical point of the energy $\int_\Sigma|p|^\alpha$, where $\alpha\in\r$. These surfaces are characterized by the Euler-Lagrange equation $H(p)=\alpha\frac{\langle N(p),p\rangle}{|p|^2}$, $p\in\Sigma$, where $H$ and $N$ are the mean curvature and the normal vector of $\Sigma$. If $\alpha\not=0$, we prove that vector planes are the only ruled $\alpha$-stationary surfaces. The second result of classification asserts that if $\alpha\not=-2,-4$, any $\alpha$-stationary surface foliated by circles must be a surface of revolution. If $\alpha=-4$, the surface is the inversion of a plane, a helicoid, a catenoid or an Riemann minimal example. If $\alpha=-2$, and besides spheres centered at $0$, we find non-spherical cyclic $-$-stationary surfaces.