Research on geometric evolution equations for hypersurfaoes

在这项研究中，我们考虑与定义在某些超曲面上的泛函相关的梯度流。梯度流是指物体向泛函梯度最陡方向的变形，是求泛函临界点的一种方法。因此，分析泛函的性质对研究具有重要意义。Helfrich变分问题是具有给定面积和封闭体积的闭合曲面之间的Willmore泛函的最小化问题。这是人类红细胞变形理论的模型之一。与之相关的梯度流称为赫尔夫里奇流。不难看出，球体是静止的溶液。长泽和Kohsaka研究了这种几何流动，得到了流动事实：(1)任意初始曲面的时间局部存在定理和唯一性定理。(2)逼近球面的初始曲面的整体存在定理。(3)Cen…的存在更多球面附近的流形及其维度的估计。这些研究结果已发表在一份学术期刊上。长泽和高木研究了从球体分叉出来的静止解。在本研究项目之前，他们已经得到了轴对称分支解的存在性和稳定性的结果。为了研究非轴对称解的存在性，导出了简化的分叉方程。此外，我们从简化的分叉方程推导出规范形，并确定了模2和4的所有解。扰动它们可能构造分叉方程的解。Sakamoto考虑了由流形的法曲率平方积分定义的泛函。他推导了第一个变分公式，并研究了临界点的结构。威尔莫尔泛函是他研究的一个特例。柳田考虑了具有三个结点的三面自由边界问题的几何流动，得到了定常解稳定的一个判据。Tchikawa从几何变分问题出发研究方程弱解的正则性。特别地，他得到了调和映射到Finsler流形的正则性的一个结果。Koike和Arisawa利用粘性解理论研究了包含几何演化方程的各种方程。Ohta和Shimokawa研究了解的爆破问题。较少

In this research, we consider gradient flows associated to functionals defined on some family of hypersurfaces. Gradient flow, which is the deformation of an object to the steepest direction of the gradient of functional, is one of methods for finding critical points of functionals. Therefore it is important for the research to analyze properties of functionals.The Helfrich variational problem is the minimizing problem of Willmore functional among the closed suefaces with the prescribed area and enclosed volume. This is one of models for shape transformation theory of human red blood cell. The associated gradient flow is called the Helfrich flow. It is not difficult to see spheres are stationary solutions. Nagasawa and Kohsaka studied this geometric flow, and obtained the flowing facts, (1)The time local existence theorem and the uniqueness theorem for arbitrary initial surfaces. (2)The global existence theorem for initial surfaces that are close to spheres. (3)The existence of the cen … More ter manifold near spheres and estimates of its dimension. These results have been submitted for an academic journal. Nagasawa and Takagi studied stationary solutions bifurcating from spheres. They had already obtained results of the existence and stability for axially symmetric bifurcating solutions before this research project. To study the existence of not necessarily axially symmetric solutions, the reduced bifurcation equation was derived. Furthermore we deduced the normal form from the reduced bifurcation equation, and determined all of solutions for modes 2 and 4. Perturbing them it might be possible to construct solutions of the bifurcation equation.Sakamoto considered the functional defined by the squared integral of normal curvature associated immersions of manifold. He derived the first variation formula and investigated the structure of critical points. The Willmore functional is a special case of his study. Yanagida considered the geometric flow associated with three-face free boundary problem with triple junction, and he got a criterion for stability of steady solutions. Tachikawa researched the regularity of weak solutions to equations from geometric variational problem. In particular he obtained a result on the regularity of harmonic maps into Finsler manifold. Koike and Arisawa considered various equations containing geometric evolution equations by using the theory of viscosity solutions. Ohta and Shimokawa researched on the blowup problem of solutions. Less