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Electronic Research Announcements

ISSN 1079-6762



Prescribing mean curvature: existence and uniqueness problems

Author: G. Kamberov
Journal: Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 4-11
MSC (1991): Primary 53C42; Secondary 35Q40, 53A05, 53A30, 53A50, 58D10, 81Q05
Published electronically: March 18, 1998
MathSciNet review: 1606327
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Abstract: 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.

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Additional Information

G. Kamberov
Affiliation: Department of Mathematics, Washington University, St. Louis, MO

Keywords: Surfaces, spinors, conformal immersions, prescribing mean curvature
Received by editor(s): February 27, 1996
Published electronically: March 18, 1998
Article copyright: © Copyright 1998 American Mathematical Society