Celebratio Mathematica

article S.-S. Chern and C. L. Terng: “An analogue of Bäcklund’s theorem in affine geometry,” Rocky Mountain J. Math. 10 : 1 (1980), pp. 105–124. MR 573866 Zbl 0407.53002

It is well known that there is a correspondence between solutions of the Sine-Gordon equation (SGE) \[ \frac{\partial^2\phi}{\partial x^2} - \frac{\partial^2\phi}{\partial t^2} = \sin\phi \] and the surfaces of constant curvature \( -1 \) in \( \mathbf{R}^3 \) (see below). The classical Bäcklund transformation of such surfaces furnishes a way to generate new solutions of the SGE from a given solution. This has received much attention in recent studies of the soliton solutions of the SGE, and the technique has been used successfully in the study of other non-linear evolution equations. In the first section of this paper we present a simple derivation of the classical Bäcklund theorem and its applications by using the method of moving frames.

Our main result concerns affine minimal surfaces. They arise as the solution of the variation problem for affine area. The corresponding Euler–Lagrange equation is a fourth order partial differential equation. In §2, we develop the basic properties of affine minimal surfaces. In §3 we study the transformation of affine surfaces by realizing them as the focal surfaces of a line congruence. The natural conditions that the congruence be a \( W \)-congruence and that the affine normals at corresponding points be parallel lead to the conclusion that both surfaces are affine minimal. This is the content of Theorem 4, the main result of our paper. As in the classical case, the Theorem leads to the construction of new affine minimal surfaces from a given one by the solution of a completely integrable system of first order partial differential equations.

incollection R. S. Palais and C.-L. Terng: “The life and mathematics of Shiing Shen Chern,” pp. 18–62 in S. S. Chern: A great geometer of the twentieth century. Edited by S.-T. Yau. Monographs in geometry and topology. International Press (Hong Kong), 1992. MR 1201340

Chuu-Lian Terng	Related
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misc J. Simons, R. Bott, R. Bryant, X.-X. Chen, S.-S. Chern, F. Hirzebruch, B. Lawson, K. Liu, E. Meinrenken, C.-L. Terng, and A. Weinstein: The Chern Symposium, 1998. Selections from MSRI’s Video Archive no. 2 (CD-ROM), recorded Berkelely, CA, 5–7 March 1998. MR 1731157 Zbl 1079.00505

F. Hirzebruch	Related	Volume
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Kefeng Liu	Related
Ekhard Meinrenken	Related

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