论文标题
威尔莫尔型能，加权区域和垂直势能的圆柱临界点之间的关系
A relation between cylindrical critical points of Willmore-type energies, weighted areas and vertical potential energies
论文作者
论文摘要
本文考虑了三种不同的物理场景的能量，并在特定情况下获得了它们之间的关系。第一个能量家族由威尔莫尔型能量组成，涉及延伸威尔莫尔和赫尔夫里奇能量的平均曲率的积分。第二个能量家族是在格罗莫夫（Gromov）开发的理论之后，当密度是高度函数的力量时，在加权歧管中产生的区域功能。第三个是当电势取决于该超平面的高度时，沉积在水平超平面中的流体的自由能。在本文中，当临界点是圆柱类型的超表面时，我们发现它们之间的关系。圆柱性超曲面取决于它们的产生平面曲线，对于每个能量家族，这些曲线满足了合适的普通微分方程。对于Willmore型能量，方程为第四顺序，而其他两种情况则为二。我们证明，在没有面积约束和加权区域的Willmore型能的生成曲线重合，并且在适当的物理参数选择后，适用于Willmore型能和垂直势能的生成曲线的相似结果。在所有情况下，生成曲线都是扩展经典弯曲能量的能量家族的关键点。在本文的最后一部分中，我们分析了沉积在具有垂直势能的水平超平面上的液滴的稳定性。事实证明，如果流体的自由界面是该超平面上的图形，那么高表面是稳定的，因为它是能量的局部最小化器。实际上，我们证明Hypererface是具有相同边界的所有图的类别中的全局最小化器。
This paper considers the energies of three different physical scenarios and obtains relations between them in a particular case. The first family of energies consists of the Willmore-type energies involving the integral of powers of the mean curvature which extends the Willmore and Helfrich energies. A second family of energies is the area functionals arising in weighted manifolds, following the theory developed by Gromov, when the density is a power of the height function. The third one is the free energies of a fluid deposited in a horizontal hyperplane when the potentials depend on the height with respect to this hyperplane. In this paper we find relations between each of them when the critical point is a hypersurface of cylindrical type. Cylindrical hypersurfaces are determined by their generating planar curves and for each of the families of energies, these curves satisfy suitable ordinary differential equations. For the Willmore-type energies, the equation is of fourth order, whereas it is of order two in the other two cases. We prove that the generating curves coincide for the Willmore-type energies without area constraint and for weighted areas, and the similar result holds for the generating curves of Willmore-type energies and of the vertical potential energies, after suitable choices of the physical parameters. In all the cases, generating curves are critical points for a family of energies extending the classical bending energy. In the final section of the paper, we analyze the stability of a liquid drop deposited on a horizontal hyperplane with vertical potential energies. It is proven that if the free interface of the fluid is a graph on this hyperplane, then the hypersurface is stable in the sense that it is a local minimizer of the energy. In fact, we prove that the hypersurface is a global minimizer in the class of all graphs with the same boundary.
