Isometric surfaces with a common mean curvature and the problem of Bonnet pairs (Q2884660)

A Bonnet pair is a pair of non-congruent surfaces in \(\mathbb R^3\) which are isometric and have the same mean curvature function. It is known that these are special surfaces, in the sense that ``most'' surfaces are determined uniquely by the metric and mean curvature. This paper studies the problem with particular attention paid to the level of differentiability involved. A formula is derived that expresses the coefficients of the second fundamental form in terms of the metric and mean curvature. Several known results are then proved under minimal smoothness assumptions for the surfaces and functions involved. Lastly it is proved that, among compact surfaces of class \(C^2\) and genus \(0\), there are no Bonnet pairs; and for \(C^2\) surfaces of non-constant mean curvature of arbitrary genus there are no Bonnet triples. There are also some sufficient conditions for the non-existence of Bonnet pairs for compact surfaces with non-constant mean curvature in the \(C^4\) class.