Research on global properties of solutions of geometric variational problems

1.研究了三维欧氏空间中具有体积约束的浸入曲面的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，能量被称为常系数椭圆参数泛函，临界点是具有常数各向异性平均曲率的曲面（CAMC曲面）。能量最小化器是一个光滑的凸曲面，称为武尔夫形状（直到平移和同构）。在各向异性表面能是旋转不变的情况下，我们得到以下结果。(1)我们证明了任何嵌入的闭CAMC曲面（直到平移和同素）是Wulff形状。(2)我们研究了某些旋转不变的椭圆参数泛函支持在两个水平面分开一个固定的距离毛细管表面。当润湿能非负时，证明了稳定解的存在唯一性。同时，我们还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的一些判据。同时，我们也得到了一些数值结果.我们得到了一个新的方法来构造CAMC曲面的例子，其表面能量不一定是旋转不变的。这不仅对CAMC表面的研究有重要意义，而且对晶体学、数学生物学等领域的研究也有重要意义.各向异性Delaunay曲面是具有常数各向异性平均曲率的旋转曲面。E表明如何生成曲线的这种表面可以获得作为跟踪的一个点的附加到一个曲线，滚动没有沿着一条线滑动。这推广了经典的常平均曲率曲面的Delaunay构造。此外，我们还利用等温自对偶性刻画了各向异性Delaunay曲面。

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.