## An Introduction to Lorentz SurfacesThe aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) |

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### Contents

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8 | |

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14 A first look at null lines | 15 |

15 The Euclidean plane E2 and the Minkowski plane E21 | 17 |

Chapter 2 Box surfaces yardsticks and global properties of Lorentzian metrics | 19 |

22 Yardsticks and timeorientability | 22 |

52 Spans on ℒ | 87 |

53 A special ℋ+ chart on the span of a null curve | 95 |

54 Characterization of C0 smoothability of the conformal boundary | 103 |

55 Kulkarnis use of the conformal boundary | 110 |

Chapter 6 Conformal invariants on Lorentz surfaces | 119 |

62 Conformal indices associated with ℒ and more properties of ℒ | 124 |

63 Some notions of symmetry | 131 |

64 Smyths digraph determining sets and some other conformal invariants | 134 |

23 Intrinsic curvature and a first look at the example in our logo | 25 |

24 Geodesics and pregeodesics | 27 |

25 Completeness inextendibility and causality conditions | 30 |

Chapter 3 Conformal equivalence and the Poincaré index | 36 |

32 Cj conformally equivalent Lorentz surfaces need not be Cj+1 conformally equivalent | 39 |

33 The Poincaré index | 47 |

34 The Poincaré Index Theorem | 50 |

Chapter 4 Kulkarnis conformal boundary | 54 |

42 The points on the conformal boundary | 61 |

43 The topology on the conformal boundary | 68 |

44 Some properties of the conformal boundary | 73 |

Chapter 5 Using the conformal boundary | 84 |

Chapter 7 Classical surface theory and harmonically immersed surfaces | 139 |

72 A quick review of local surface theory in Minkowski 3space | 149 |

73 Contrasting the behavior of surfaces in E3 and E31 | 162 |

74 The HilbertHolmgren theorem for harmonically immersed surfaces | 166 |

Chapter 8 Conformal realization of Lorentz surfaces in Minkowski 3space | 175 |

82 Associate families of minimal surfaces | 182 |

83 Some conformal realizations of Lorentz surfaces in E31 | 188 |

84 Some last remarks on conformal imbeddings and immersions | 196 |

Bibliography | 201 |

205 | |