Research on global properties of solutions of geometric variational problems

1. 研究了具有体积约束的欧几里得三维浸没曲面各向异性表面能的临界点。我们假设表面能满足一定的“凸性条件”。在这种情况下，能量称为常系数椭圆参数泛函，临界点是具有恒定各向异性平均曲率的曲面（CAMC曲面）。能量最小化是一个光滑的凸表面，称为伍尔夫形状（直到平移和同质）。在各向异性表面能是旋转不变的情况下，我们得到了以下结果。(1)我们证明了任何嵌入的封闭CAMC曲面都是（在平移和同质条件下）Wulff形状。(2)研究了若干旋转不变椭圆参数泛函的毛细曲面，这些泛函支撑在两个固定距离的水平面上。当润湿能为非负时，证明了稳定解的存在唯一性。同时，我们确定了解的几何性质。当润湿能为负时，得到了稳定解存在唯一性的判据。并对该问题进行了数值计算。我们得到了一种构造表面能不一定是旋转不变的CAMC曲面实例的新方法。该方法不仅可用于CAMC表面的研究，还可用于晶体学、数学生物学等其他领域的研究。各向异性德劳奈曲面是具有恒定各向异性平均曲率的旋转曲面。E展示了如何将这些曲面的生成曲线作为一条曲线上的点的轨迹，这条曲线沿着一条直线滚动而不滑动。这推广了由Delaunay引起的常平均曲率曲面的经典构造。此外，我们利用各向异性德劳奈曲面的等温自对偶性来表征它们。

1. We studied critical points of an anisotropic surface energy for immersed surfaces in the euclidean three-space with a volume constraint. We assume that the surface energy satisfies a certain "convexity condition". In this case, the energy is called a constant coefficient elliptic parametric functional, and the critical points are surfaces with constant anisotropic mean curvature (CAMC surfaces). The energy-minimizer is a smooth convex surface which is called the Wulff shape (up to translation and homothety). In the case where the anisotropic surface energy is rotationally invariant, we obtained the following results.(1) We proved that any embedded closed CAMC surface was (up to translation and homothety) the Wulff shape.(2) We studied capillary surfaces for certain rotationally invariant elliptic parametric functionals supported on two horizontal planes separated by a fixed distance. When the wetting energy is nonnegative, we proved the existence and uniqueness of the stable solution. Also, we determined the geometric property of the solution. When the wetting energy is negative, we obtained some criterions for the existence and uniqueness of the stable solution. Also, we obtained some numerical results on this problem.2. We obtained a new method of constructing examples of CAMC surfaces whose surface energy is not necessarily rotationally invariant. It will be useful not only for the study on CAMC surfaces but also for other fields such as crystallography, mathematical biology, and so on.3. Anisotropic Delaunay surfaces are surfaces of revolution with constant anisotropic mean curvature. e showed how the generating curves of such surfaces could be obtained as the trace of a point attached to a curve which was rolled without slipping along a line. This generalizes the classical construction for constant mean curvature surfaces due to Delaunay. Moreover, we characterize anisotropic Delaunay surfaces by using their isothermic self-duality.