Analysis of gradient flow for the bending energy of plane curves under multiple constraints

Helfrich变分问题是在长度和封闭体积约束下的封闭平面曲线的弯曲能问题。该问题不仅适用于曲线，也适用于曲面和超曲面，即高维情况。我们研究了相应的梯度流的行为，称为“富helrich流”。首先构造一般维的helrich流，并讨论其唯一性。此外，我们还得到了广义旋转超曲面的整体存在性的一个新事实，这对分析富helrich流的行为是有帮助的。

The Helfrich variational problem is one for the bending energy of closed plane curves under constrains of length and enclosed volume. The problem can be formulated not only for curves but also for surfaces and hypersurface, i.e., higher dimensional case. We study the behavior of the corresponding gradient flow, called the “Helfrich flow". Firstly, we construct the Helfrich flow of general dimension, and discuss the uniqueness. Furthermore we get a new fact on the global existence of generalized rotational hypersurface which is useful for the analysis of behavior for the Helfrich flow.