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

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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.