Functinal analytic approach to reseaches on deformation and stability of surfaces with constant mean curvature

1.研究了平均曲率为水平面上方高度的线性函数的曲面(PMC曲面)的变分理论。我们发展了一个通量公式，并用它证明了闭PMC曲面的不存在性结果。研究了PMC曲面的摄动理论。我们得到了具有平面边界的PMC曲面稳定的必要条件。2.提出了一个变分问题，该问题的解是部分由给定轮廓支撑的有重力薄膜的几何模型。能量泛函包含表面张力、引力能和润湿能，欧拉-拉格朗日方程可以用表面的平均曲率、自由边界的曲率和其他几个几何量来表示。特别是，我们详细地研究了一个简单的情况，其中解是以两条垂直线为界的垂直平面。我们确定了每个解的稳定性或不稳定性。3.我们研究了具有体积约束的各向异性表面能量处于平衡状态的曲面的几何。我们得到了这类曲面的一阶和二阶变分，并研究了这类曲面的Gauss映射的例外集。我们还得到了平衡旋转曲面的表示式，并研究了这些曲面的几何和稳定性。

1.We studied the variational theory of surfaces whose mean curvature is prescribed to be a linear function of their height above a horizontal plane (PMC surfaces). We developed a flux formula and used it to prove nonexistence results for closed PMC surfaces. The perturbation theory for PMC surfaces was studied. We obtained necessary conditions for the stability of PMC surfaces with planar boundaries.2.We posed a variational problem for surfaces whose solutions are a geometric model for thin films with gravity which is partially supported by a given contour. The energy functional contains surface tension, a gravitational energy and a wetting energy, and the Euler-Lagrange equation can be expressed in terms of the mean curvature of the surface, the curvatures of the free boundary and a few other geometric quantities. Especially, we studied in detail a simple case where the solutions were vertical planar surfaces bounded by two vertical lines. We determined the stability or instability of each solution.3.We studied the geometry of surfaces which are in equilibrium for an anisotropic surface energy with a volume constraint. We obtained the first and second variations and studied the exceptional set of the Gauss map for such surfaces. Also we obtained representation formulas of the equilibrium surfaces of revolution and studied the geometry and stability of these surfaces.