Painleve Equations in the Differential Geometry of Surfaces

1 Introduction 1 2. Basics on Painlevé Equations and Quaternionic Description of Surfaces 7 2.1 Painleve Property and Painleve Equations 7 2.2 Isomonodromic Deformations 9 2.3 Conformaily Parametrized Surfaces 16 2.4 Quaternionic Description of Surfaces 19 3. Bonnet Surfaces in Euclidean Three-space 21 3.1 Definition of Bonnet Surfaces and Simplest Properties 22 3.2 Local Theory away from Critical Points 26 3.3 Local Theory at Critical Points 33 3.3.1 Index of a Critical Point 34 3.3.2 Hazzidakis Equation at Critical Points 35 3.3.3 Connection to the Local Theory away from Critical Points 39 3.4 Bonnet Surfaces via Painleve T'ranscendents 39 3.4.1 Prom the Moving Frame of Bonnet B and B_nu Surfaces to the Lax Representation of Painlevé VI Equations 40 3.42 Moving Frame Equation and Lax Representation for Painlevé V 44 3.5 Global Properties of Bonnet Surfaces 49 3.5.1 Existence of Critical Points 49 3.5.2 Global Classification of Bonnet Surfaces 51 3.6 Examples of Bonnet Surfaces 54 3.6.1 Bonnet Surface of type A, B, and C 54 3.6.2 Bonnet Surfaces with Critical Points 56 3.7 Schlesinger Transformations for Bonnet Surfaces. 58 3.7.1 Transformations of Painleve Equations 58 3.7.2 Schlesinger Transformations for Bonnet Surfaces of Type B 59 3.7.3 Schlesinger Transformation for Bonnet Surfaces with Critical Points 62 4. Bonnet Surfaces in S^3 and H^3 and Surfaces with HarmonicInverse Mean Curvature 65 4.1 Surfaces in S^3 and H^3 65 4.2 Definition and Simplest Properties 67 4.3 Bonnet Surfaces in S^3 and H^3 away from Critical Points 68 4.4 Local Theory of Bonnet Surfaces in S^3 and H^3 at Critical Points 71 4.5 Bonnet surfaces in S^3 and H^3 in Terms of Painleve Transcendents 72 4.6 Global Properties of Bonnet Surfaces in Space Forms 74 4.7 Surfaces with Harmonic Inverse Mean Curvature 75 4.8 Bonnet Pairs of HIMC Surfaces 78 4.8.1 Basic Facts about Bonnet Pairs 78 4.8.2 Bonnet Surfaces in S^3 and Bonnet Pairs of HIMC Surfaces 80 HIMC Bonnet Pairs in Painlevé Tanscendents 83 4.9.1 HIMC Bonnet Pairs of type A and Painlevé VI Equations 83 4.9.2 HIMC Bonnet Pairs of type C and Painlevé V Equations 85 4.10 Examples of HIMC Surfaces 87 5. Surfaces with Constant Curvature 89 5.1 Surfaces with Constant Negative Gaussian Curvature and Two Straight Asymptotic Lines 89 5.1.1 Surfaces with Constant Negative Gaussian Curvature 89 51.2 Amsler Surfaces 91 5.1.3 The asymptotic cone and self-similar evolution of smoke-rings 94 5.2 Smyth surfaces 98 5.3 Affine Spheres with Affine Straight Lines 101 53.1 Indefinite Aifine Spheres 101 5.3.2 Curves in Affine Differential Geometry and Asymptotic Lines on Affine Spheres 103 5.3.3 Affine Spheres with Affine Straight Lines and Painlevé III Equation 105 5.3.4 Examples of Affine Spheres 107 6. Appendices lO9 6.1 Appendix A. Proof of Lemma 3.3.2: Non-existence of Umbilic Points with M = 0 109 6.2 Appendix B. Proof of Lemma 3.5.1: Existence of Critical Points 111 References 112 Index 119 END.