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An Introduction to Lorentz Surfaces
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Table of contents
- Introduction
- Chapter 1. Null lines on Lorentz surfaces
- § 1.1. Scalar products and causal character
- § 1.2. Metrics and null direction fields
- § 1.3. Lorentz surfaces and proper null coordinates
- § 1.4. A first look at null lines
- § 1.5. The Euclidean plane E2 and the Minkowski plane E21
- Chapter 2. Box surfaces, yardsticks and global properties of Lorentzian metrics
- § 2.1. The one-one correspondence between box surfaces and Lorentz surfaces
- § 2.2. Yardsticks and time-orientability
- § 2.3. Intrinsic curvature and a first look at the example in our logo
- § 2.4. Geodesics and pregeodesics
- § 2.5. Completeness, inextendibility, and causality conditions
- Chapter 3. Conformal equivalence and the Poincaré index
- § 3.1. Definitions of conformal equivalence
- § 3.2. Cj conformally equivalent Lorentz surfaces need not be Cj+1 conformally equivalent
- § 3.3. The Poincaré index
- § 3.4. The Poincaré Index Theorem
- Chapter 4 Kulkarni’s conformal boundary
- § 4.1. Ideal endpoints
- § 4.2. The points on the conformal boundary
- § 4.3. The topology on the conformal boundary
- § 4.4. Some properties of the conformal boundary
- Chapter 5 Using the conformal boundary
- § 5.1. The foliations X and Y
- § 5.2. Spans on ℒ
- § 5.3. A special ℋ+ chart on the span of a null curve
- § 5.4. Characterization of C0 smoothability of the conformal boundary
- § 5.5. Kulkarni’s use of the conformal boundary
- Chapter 6. Conformal invariants on Lorentz surfaces
- § 6.1. Conformal indices on an arbitrary Lorentz surface
- § 6.2. Conformal indices associated with ∂ℒ and more properties of ∂ℒ
- § 6.3. Some notions of symmetry
- § 6.4. Smyth’s digraph, determining sets and some other conformal invariants
- Chapter 7. Classical surface theory and harmonically immersed surfaces
- § 7.1. A quick review of local surface theory in Euclidean 3-space
- § 7.2. A quick review of local surface theory in Minkowski 3-space
- § 7.3. Contrasting the behavior of surfaces in E3 and E3,1
- § 7.4. The Hilbert-Holmgren theorem for harmonically immersed surfaces
- Chapter 8. Conformal realization of Lorentz surfaces in Minkowski 3-space
- § 8.1. Entire timelike minimal surfaces in E3,1
- § 8.2. Associate families of minimal surfaces
- § 8.3. Some conformal realizations of Lorentz surfaces in E3,1
- § 8.4. Some last remarks on conformal imbeddings and immersions
- Bibliography
- Index