Title: Some Applications of Surface Curvatures in Theoretical Physics Show Chinese title

Title: 曲面曲率在理论物理中的某些应用

Abstract: In this survey article, we present two applications of surface curvatures in theoretical physics. The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy. In this formalism, the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics. We first show that there is an obstruction, arising from the spontaneous curvature, to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori. We then propose a scale-invariant anisotropic bending energy, which extends the Canham energy, and show that it possesses a unique toroidal energy minimizer, up to rescaling, in all parameter regime. Furthermore, we establish some genus-dependent topological lower and upper bounds, which are known to be lacking with the Helfrich energy, for the proposed energy. We also present the shape equation in our context, which extends the Helfrich shape equation. The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings. In this formalism, gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector. This setting provides a lucid exhibition of the interplay of the underlying geometry, matter energy, and topological characterization of the system. In both areas of applications, we encounter highly challenging nonlinear partial differential equation problems. We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered. Show Chinese abstract

Abstract: 在本文综述文章中，我们介绍了曲面曲率在理论物理中的两个应用。 第一个应用源于生物物理学，在研究细胞囊泡形状时涉及一种称为赫尔弗里奇弯曲能的平均曲率类型能量的最小化。 在此形式中，细胞囊泡的平衡形状可能呈现出丰富的几何和拓扑特征。 我们首先表明，由于自发曲率的存在，在嵌入环面的集合上，赫尔弗里奇能量的极小值可能存在障碍。 然后我们提出了一种尺度不变的各向异性弯曲能，它扩展了坎姆能量，并证明在所有参数范围内，该能量在尺度变换下具有唯一的环面能量极小值。 此外，我们建立了一些与种类相关的拓扑下界和上界，这些界限在赫尔弗里奇能量中是缺乏的，适用于所提出的能量。 我们还介绍了我们上下文中的形状方程，它扩展了赫尔弗里奇形状方程。 第二个应用源于天体物理学，在宇宙弦的背景下寻找早期宇宙中物质吸积的机制。 在此形式中，引力可能仅存储在一个二维曲面上，使得爱因斯坦张量用该曲面的高斯曲率表示，该曲率直接与物质部分的哈密顿能量密度相关。 这种设置提供了对系统底层几何、物质能量和拓扑特征之间相互作用的清晰展示。 在这些应用领域中，我们遇到了高度复杂的非线性偏微分方程问题。 我们证明，对这些方程的研究有助于我们理解所考虑的理论物理问题。