Loading...
Please wait, while we are loading the content...
Similar Documents
And the Analytic Reflection Principle (part I) Chapter 1: General Introduction for Part I §1.1. Resumed Historical Background
| Content Provider | Semantic Scholar |
|---|---|
| Author | Merker, Jo¨el |
| Copyright Year | 2004 |
| Abstract | 1.1.1. Local Lie groups and the no Riemann mapping theorem at the boundary. Inspired by the general idea that, in analogy withÉ. Galois's group theory of algebraic equations, group analysis of differential equations should provide precious information about their solvability, S. Lie began around 1873–80 the classification of all continuous local groups of transformation acting on C n. He quickly succeeded for n = 1 and achieved the case n = 2 (see [18]), but the unavoidable complexity and richness for n = 3 exhausted his efforts; moreover, after more than one century, the task has never been achieved. Nevertheless, especially for n = 2, Lie's classificationf had the enormous power of providing any possible application to the study of transformations preserving arbitrary types of geometric structures. Thanks to the influence of G. Darboux, the works of S. Lie were rapidly known to French mathematicians. Based on the general approach of S. Lie, H. Poincaré (see [24]) discovered in 1907 that the automorphism groups of the two-dimensional unit ball B 2 := {(z 1 , z 2) ∈ C 2 : |z 1 | 2 + |z 2 | 2 < 1} and of the bidisc ∆ 2 := {(z 1 , z 2) ∈ C 2 : |z 1 | < 1, |z 2 | < 1} are represented by rational, but not isomorphic transformations and he deduced immediately that B 2 and ∆ 2 are not biholomorphically equivalent. This discovery was the starting point of the no Riemann mapping theorem in several complex variables. In the beginning of the twentieth century, the birth of pluricomplex geometry also coincided with two other fundamental memoirs of F. culminated in the complete solution of the so-called problem of Levi given by K. Oka in 1951–52, the direction initiated by H. Poincaré in 1907 lay dormant for approximatively sixty years, with the major exception of four consecutive and historically isolated memoirs of B. Segre [25], [26] and ofÉ. Cartan [3], [4] in the years 1931-32. Based on works of S. Lie, of A. Tresse (a french student of S. Lie), and of the young mathematician B. Segre, ´ E. Cartan (who also had defended his thesis under the direction of S. Lie) provided an essentially complete classification of all Levi nondegenerate real analytic local hypersurfaces in C 2 , which ultimately relies on S. Lie's far reaching works about the classification of … |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/math/0404249v1.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
