Matematiska institutionen

Draft of the book on extremal length (joint with Lars Ahlfors)

Content of the draft:

Version A

Chap.I. Quelques notions et méthodes de l’analyse réelle.

§1 Fonctions à carré intégrable 1–4
§2 Fonctions harmoniques et sous-harmoniques  4bis-11
§3 Le problème de Dirichlet 12-19

Chap.II. Propriétés extrémales de représentations conformes.

§1 Représentation circulaire 1–13
§2 Longeurs et distances extrémales 16–23
§3 La correspondence des frontières 24–28

Version B

End of the section with discussion of capacity  56-57
§2 Longeurs et distances extrémales 58-66
§3 La correspondence des frontières 67-70
§4 Détermination de la métrique extrémale 71–76
§5 Les lois de composition  77–84
§6 La distance extrémale dans le cas de points isolés 85–102
§7 Applications  103–112
§8 La transportation des masses 113–126

Additional parts

§1 Le quadrilatère  1–6
§2 Le pentagone  7–20

It seems that at least two different versions of the manuscript existed. What remained of the first version consists of Chap. I and the first three sections of Chapter II.

Each section has a new numeration of the pages. The second version starts with discussion of capacity, from which only two pages (56–57) survived and then covers sections 2–8 of Chap. II.

The additional parts look close to the Ahlfors talk at the 10th Scandinavian Math. Congress at Copenhagen, 1946. Some figures in this part are hardly visible.

According to this description, we rearranged the material into three files.

The location of the original manuscripts remains unknown to us. We have not seen them, and we worked only with their copies in pdf format.