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Since the time of surfaces -+ in differential Gauss, parametrized (x, y) P(x, y) have been described a frame attached to the moving geometry through TI(x, y) surface. One introduces the Gauss- which linear dif- Weingarten equations are , ferential equations = U = TIX T1, VT', !PY (1. for the and their condition frame, compatibility - = V + [U, V] 0, UY (1.2) which the Gauss-Codazzi For surfaces in three-dim- represents equations . a sional Euclidean the frame T1 lies in the usually or space, group SO(3) SU(2). On the other a of a non-linear in the form hand, representation equation (1.2) is the of the of of starting point theory integrable equations (theory solitons), which in mathematical in the 1960's appeared physics [NMPZ, AbS, CD, FT, More the differential for the coefficients of AbC]. exactly, partial equation (1.2) the matrices U and V is considered to be if these matrices can be integrable , extended to U V non-trivially a one-parameter family (x, y, A), (x, y, A) satisfying - = + U(A)y V(A). [U(A), V(A)] 0, (1-3) so that the differential is and original partial equation preserved.' . Usually U(A) V are rational functions of the which is called the (A) parameter A, spectral param- In soliton the eter is called the Lax . theory, representation (1.3) representation the Zakharov-Shabat or representation [ZS].

Painleve Equations in the Differential Geometry of Surfaces

1. Introduction 2. Basics of Painlevé Equations and Quaternionic Description of Surfaces2.1. Painlevé Property and Painlevé Equations2.2. Isomonodromic Deformations 2.3. Conformally Parametrized Surfaces 2.4. Quaternionic Description of Surfaces 3. Bonnet Surfaces in Euclidean three-space 3.1. Definition of Bonnet Surfaces and Simplest Properties 3.2. Local Theory away from Critical Points 3.3. Local Theory at Critical Points 3.4. Bonnet Surfaces via Painlev Transcendents 3.5. Global Properties of Bonnet Surfaces 3.6. Examples of Bonnet Surfaces 3.7. Schlesinger Transformations for Bonnet Surfaces 4. Bonnet Surfaces in S and H and Surfaces with Harmonic Inverse Mean Curvature 4.1. Surfaces in S3 and H3 4.2. Definition and Simplest Properties 4.3. Bonnet Surfaces in S3 and H3 away from Critical Points 4.4. Local Theory of Bonnet Surfaces in S and H at Critical Points 4.5. Bonnet Surfaces in S3 and H3 in Terms of Painlev Transcendents 4.6. Global Properties of Bonnet Surfaces in Space Forms 4.7. Surfaces with Harmonic Inverse Mean Curvature 4.8. Bonnet Pairs of HIMC Surfaces 4.9. HIMC Bonnet Pairs in Painlev Transcendents 4.10. Examples of HIMC Surfaces 5. Surfaces with Constant Curvature 5.1. Surfaces with Constant Negative Gaussian Curvature and Two Straight Asymptotic Lines 5.2. Smyth Surfaces 5.3. Affine Spheres with Affine Straight Lines 6. Appendices