Parabolic Riemannian Planes Carrying Biharmonic Green's Functions of the Clamped Plate.

摘要

The existence on a Riemannian manifold of a biharmonic Green's function gamma of a simply supported body, with boundary data gamma = delta gamma = 0, is known to entail the existence of both the biharmonic Green's function beta of a clamped body, with boundary data beta = the normal derivitive of beta = 0, and the harmonic Green's function g. Regarding relations between the existence of beta and g, it is known that there are Riemannian manifolds which carry neither beta nor g (2-space), g but not beta (3-space), and both beta and g (5-space). Whether or not there exist manifolds which carry beta but no g has been an open question. The main purpose of the present study is to solve this problem by showing that there do exist Riemannian manifolds which are parabolic, i.e., carry no g, but nevertheless carry beta.