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

DOI:
https://doi.org/10.1090/S1079-6762-98-00040-7

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 $\mathbb {R}^{3}$ or its shape. The emphasis is on Bonnet’s problem: classify and study the surface immersions in $\mathbb {R}^3$ 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.

- Michael F. Atiyah,
*Riemann surfaces and spin structures*, Ann. Sci. École Norm. Sup. (4) **4** (1971), 47–62. MR **286136**, DOI 10.24033/asens.1205
- Allan P. Fordy and John C. Wood (eds.),
*Harmonic maps and integrable systems*, Aspects of Mathematics, E23, Friedr. Vieweg & Sohn, Braunschweig, 1994. MR **1264179**, DOI 10.1007/978-3-663-14092-4
- O. Bonnet,
*Mémoire sur la théorie des surfaces applicables*, J. Éc. Polyt. **42** (1867).
- Albert Eagle,
*Series for all the roots of a trinomial equation*, Amer. Math. Monthly **46** (1939), 422–425. MR **5**, DOI 10.2307/2303036
- Shiing Shen Chern,
*Deformation of surfaces preserving principal curvatures*, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 155–163. MR **780041**
- Robert B. Gardner,
*An integral formula for immersions in euclidean space*, J. Differential Geometry **3** (1969), 245–252. MR **254785**
- Nigel Hitchin,
*Harmonic spinors*, Advances in Math. **14** (1974), 1–55. MR **358873**, DOI 10.1016/0001-8708(74)90021-8
- Heinz Hopf,
*Differential geometry in the large*, Lecture Notes in Mathematics, vol. 1000, Springer-Verlag, Berlin, 1983. Notes taken by Peter Lax and John Gray; With a preface by S. S. Chern. MR **707850**, DOI 10.1007/978-3-662-21563-0
- G. Kamberov,
*Recovering the shape of a surface from the mean curvature*, to appear in *Proc. of the Intl. Conf. on Differential Equations and Dynamical Systems*, 1996, publ. by Marcel Dekker, (1998).
- G. Kamberov,
*Lectures on quaternions, spinors, and surfaces*, Notes from the lectures at the University of São Paulo, Brazil, May–June, 1997.
- G. Kamberov,
*Quadratic differentials, quaternionic forms, and surfaces*, preprint.
- G. Kamberov,
*Immersions of closed surfaces with prescribed mean curvature half-density*, in preparation.
- G. Kamberov,
*The Dirichlet problem for Dirac spinors*, preprint.
- Katsuei Kenmotsu,
*The Weierstrass formula for surfaces of prescribed mean curvature*, Minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977) North-Holland, Amsterdam-New York, 1979, pp. 73–76. MR **574254**
- G. Kamberov, P. Norman, F. Pedit, U. Pinkall,
*Surfaces, quaternions, and spinors*, preprint.
- G. Kamberov, F. Pedit, and U. Pinkall,
*Bonnet pairs and isothermic surfaces*, GANG Preprint IV.21, 1996, to appear in Duke Math. J. **92** (1998).
- B. G. Konopelchenko,
*Induced surfaces and their integrable dynamics*, Stud. Appl. Math. **96** (1996), no. 1, 9–51. MR **1365273**, DOI 10.1002/sapm19969619
- B. G. Konopelchenko and I. A. Taimanov,
*Generalized Weierstrass formulae, soliton equations and the Willmore surfaces, I. Tori of revolution and the mKDV*, preprint no. 187, Ruhr-Universität-Bochum, Fakultät für Mathematik, 1995.
- R. Kusner and N. Schmidt,
*The spinor representation of minimal surfaces*, GANG Preprint III.27, 1993. Revised version, GANG Preprint 4.18.
- H. Blaine Lawson Jr. and Renato de Azevedo Tribuzy,
*On the mean curvature function for compact surfaces*, J. Differential Geometry **16** (1981), no. 2, 179–183. MR **638784**
- F. Pedit,
*Lectures at the Summer School in Differential Geometry*, Durham, UK, Summer 1996.
- U. Pinkall,
*Regular homotopy classes of immersed surfaces*, Topology **24** (1985), no. 4, 421–434. MR **816523**, DOI 10.1016/0040-9383(85)90013-8
- U. Pinkall,
*Lectures at TU Berlin*, Spring 1996.
- J. Richter,
*Surfaces in terms of spinors*, Master Thesis, TU Berlin, 1995.
- Antonio Ros,
*Compact hypersurfaces with constant scalar curvature and a congruence theorem*, J. Differential Geom. **27** (1988), no. 2, 215–223. With an appendix by Nicholas J. Korevaar. MR **925120**
- D. Sullivan,
*The spinor representation for minimal surfaces in space*, Unpublished notes, 1989.

- M. Atiyah,
*Riemann surfaces and spin structures*, Ann. Sci. Ec. Norm. Sup. **4** (1971), 47–62.
- A. I. Bobenko,
*Surfaces in terms of 2 by 2 matrices*, Harmonic Maps and Integrable Systems (A. Froddy and J. Wood, eds.), Aspects of Mathematics, Vieweg, Braunschweig, 1994.
- O. Bonnet,
*Mémoire sur la théorie des surfaces applicables*, J. Éc. Polyt. **42** (1867).
- É. Cartan,
*Sur les couples de surfaces applicables avec conservation des courbures principales*, Bull. Sc. Math. **66** (1942), 55–72, 74–85.
- S. S. Chern,
*Deformations of surfaces preserving principal curvatures*, Differential Geometry and Complex Analysis: a Volume Dedicated to the Memory of H. E. Rauch (I. Chavel and H. M. Farkas, eds.), Springer-Verlag, 1985, pp. 155–163.
- R. Gardner,
*An integral formula for immersions in Euclidean space*, J. Diff. Geom. **3** (1969), 245–252.
- N. Hitchin,
*Harmonic spinors*, Adv. in Math. **14** (1974), 1–55.
- H. Hopf,
*Differential geometry in the large*, Lecture Notes in Mathematics, vol. 1000, Springer-Verlag, Berlin and New York, 1983.
- G. Kamberov,
*Recovering the shape of a surface from the mean curvature*, to appear in *Proc. of the Intl. Conf. on Differential Equations and Dynamical Systems*, 1996, publ. by Marcel Dekker, (1998).
- G. Kamberov,
*Lectures on quaternions, spinors, and surfaces*, Notes from the lectures at the University of São Paulo, Brazil, May–June, 1997.
- G. Kamberov,
*Quadratic differentials, quaternionic forms, and surfaces*, preprint.
- G. Kamberov,
*Immersions of closed surfaces with prescribed mean curvature half-density*, in preparation.
- G. Kamberov,
*The Dirichlet problem for Dirac spinors*, preprint.
- K. Kenmotsu,
*Weierstrass formula for surfaces of prescribed mean curvature*, Math. Ann. **245** (1979), 89–99.
- G. Kamberov, P. Norman, F. Pedit, U. Pinkall,
*Surfaces, quaternions, and spinors*, preprint.
- G. Kamberov, F. Pedit, and U. Pinkall,
*Bonnet pairs and isothermic surfaces*, GANG Preprint IV.21, 1996, to appear in Duke Math. J. **92** (1998).
- B. G. Konopelchenko,
*Induced surfaces and their integrable dynamics*, Stud. Appl. Math. **96** (1996), 9–51.
- B. G. Konopelchenko and I. A. Taimanov,
*Generalized Weierstrass formulae, soliton equations and the Willmore surfaces, I. Tori of revolution and the mKDV*, preprint no. 187, Ruhr-Universität-Bochum, Fakultät für Mathematik, 1995.
- R. Kusner and N. Schmidt,
*The spinor representation of minimal surfaces*, GANG Preprint III.27, 1993. Revised version, GANG Preprint 4.18.
- B. Lawson and R. Tribuzy,
*On the mean curvature function for compact surfaces*, J. Diff. Geom. **16** (1981), 179–183.
- F. Pedit,
*Lectures at the Summer School in Differential Geometry*, Durham, UK, Summer 1996.
- U. Pinkall,
*Regular homotopy classes of immersed surfaces*, Topology **24** (1985), 421–435.
- U. Pinkall,
*Lectures at TU Berlin*, Spring 1996.
- J. Richter,
*Surfaces in terms of spinors*, Master Thesis, TU Berlin, 1995.
- A. Ros,
*Compact hypersurfaces with constant scalar curvature and a congruence theorem*, J. Diff. Geom. **27** (1988), 215–223.
- D. Sullivan,
*The spinor representation for minimal surfaces in space*, Unpublished notes, 1989.

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Additional Information

**G. Kamberov**

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

Email:
kamberov@math.wustl.edu

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