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The paper presents results on the extent to which mean curvature data can be used to determine a surface in $\bR^{3}$ or its shape. The emphasis is on Bonnet's problem: classify and study the surface immersions in $\bR^3$ 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.

Copyright 1998 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 04 (1998), pp. 4-11
• Publisher Identifier: S 1079-6762(98)00040-7
• 1991 Mathematics Subject Classification. Primary 53C42; Secondary 35Q40, 53A05, 53A30, 53A50, 58D10, 81Q05
• Key words and phrases. Surfaces, spinors, conformal immersions, prescribing mean curvature
• Received by the editors February 27, 1996
• Posted on March 18, 1998
• Comments (When Available)

G. Kamberov

Department of Mathematics, Washington University, St. Louis, MO

E-mail address: kamberov@math.wustl.edu