Prescribing mean curvature: existence and uniqueness problems

The paper presents results on the extent to which mean curvature data can be used to determine a surface in $\mathbf{R}^{3}$ or its shape. The emphasis is on Bonnet's problem: classify and study the surface immersions in $\mathbf{R}^3$ whose shape is not uniquely determined by the first fundamental form and the mean curvature function. These immersions are called Bonnet immersions. A local solution of Bonnet's problem for umbilic-free immersions follows from papers by Bonnet, Cartan, and Chern. The properties of immersions with umbilics and global rigidity results for closed surfaces are presented in the first part of this paper. The second part of the paper outlines an existence theory for conformal immersions based on Dirac spinors along with its immediate applications to Bonnet's problem. The presented existence paradigm provides insight into the topology of the moduli space of Bonnet immersions of a closed surface, and reveals a parallel between Bonnet's problem and Pauli's exclusion principle.