Extensions through codimension one to sense preserving mappings

Titus, Charles J.

Annales de l'Institut Fourier, Tome 23 (1973) no. 2, pp. 215-227.

L’archétype des questions considérées ici est le suivant : “Quelles sont les courbes planes orientées qui peuvent être représentées comme images de la frontière d’un disque par une fonction holomorphe ?” Cette question est équivalente à la suivante : “Quelles sont les immersions du cercle dans le plan possédant une extension au disque fermé régulière, préservant l’orientation ($\equiv$ à jacobien non négatif)” ?

La seconde question est généralisée en termes du genre et de la dimension des ensembles de départ et d’arrivée. L’exposé est fait en termes de la motivation, des résultats, des méthodes et des conjectures.

The archetype for the questions considered is: “Which plane oriented curves in the plane are representable as the images of the boundary of a disk under holomorphic function?” This question is equivalent to: “Which immersion of the circle in the plane are extendable to smooth sense preserving (= non-negative jacobian) mappings of the closed disk with the jacobian positive on the boundary?”

The second question is generalized in terms of the genus and dimension of the source and target. An exposition is given in terms of motivation, results, approaches and conjectures.

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