Research on geometric evolution equations for hypersurfaoes

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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