Research on geometric evolution equations for hypersurfaoes

在这项研究中，我们考虑梯度流定义在一些家庭的超曲面泛函。梯度流是求泛函临界点的方法之一，它是物体向泛函梯度最陡方向的变形。Helfrich变分问题是Willmore泛函在具有给定面积和封闭体积的闭曲面之间的极小化问题。这是人体红细胞形态转化理论的模型之一。相关的梯度流称为Helfrich流。不难看出球面是定常解。Nagasawa和Kohsawa研究了这种几何流，得到了流动的事实：（1）任意初始曲面的时间局部存在性定理和唯一性定理。（2）逼近球面的初始曲面的整体存在性定理。（3）中心的存在 ...更多信息 称之为近球流形及其维数估计。这些结果已提交给一份学术期刊。Nagasawa和Takagi研究了从球体分叉的固定解。在此之前，他们已经得到了轴对称分支解的存在性和稳定性的结果。为了研究不一定轴对称解的存在性，导出了约化分歧方程。进而由约化分岔方程推导出规范形，确定了模态2和模态4的所有解。Sakamoto考虑了由流形的法曲率的平方积分所定义的泛函，并讨论了流形的浸入问题。他推导出第一变分公式，并研究了临界点的结构。Willmore函数是他研究的一个特例。Yanagida考虑了与三重连接的三面自由边界问题有关的几何流，得到了定常解稳定性的一个判据。Tachikawa从几何变分问题出发研究了方程弱解的正则性。特别是他获得了一个结果的规律性调和映射到芬斯勒流形。小池和有泽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。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