Metrics and convergence in moduli spaces of maps

Joseph Palmer

doi:10.5802/afst.1577

Joseph Palmer ¹

¹ University of California, San Diego, Mathematics Department, 9500 Gilman Drive, #0112, La Jolla, CA 92093-0112, USA.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 3, pp. 497-526.

Résumé
Abstract

Nous présentons un cadre général pour l’étude de la convergence des familles d’applications entre variétés dont les domaines de définition sont distincts. Étant données deux variétés $M$ et $N$ , $M$ étant munie d’une forme volume, nous considérons des familles d’applications dans l’ensemble ${(φ, B_{φ}) ∣ B_{φ} \subset M, φ : B_{φ} \to N avec B_{φ}, φ mesurable}$ et nous définissons une distance sur cet ensemble, de type distance $L^{1}$ généralisée. Nous démontrons que l’espace métrique ainsi obtenu est toujours complet. Nous nous concentrons ensuite sur l’étude des propriétés de convergence de telles familles d’applications.

We provide a general framework to study convergence properties of families of maps between manifolds which have distinct domains. For manifolds $M$ and $N$ where $M$ is equipped with a volume form we consider families of maps in the collection ${(φ, B_{φ}) ∣ B_{φ} \subset M, φ : B_{φ} \to N with B_{φ}, φ both measurable}$ and we define a distance function similar to a generalized $L^{1}$ distance on such a collection. We show that the resulting metric space is always complete. We then shift our focus to exploring the convergence properties of families of such maps.

Reçu le : 2016-03-12
Accepté le : 2017-09-24
Publié le : 2018-09-19

MR Zbl

DOI : 10.5802/afst.1577

Affiliations des auteurs :

Joseph Palmer ¹

¹ University of California, San Diego, Mathematics Department, 9500 Gilman Drive, #0112, La Jolla, CA 92093-0112, USA.

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Joseph Palmer},
     title = {Metrics and convergence in moduli spaces of maps},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {497--526},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {3},
     year = {2018},
     doi = {10.5802/afst.1577},
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     zbl = {1404.58020},
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Joseph Palmer. Metrics and convergence in moduli spaces of maps. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 3, pp. 497-526. doi : 10.5802/afst.1577. https://afst.centre-mersenne.org/articles/10.5802/afst.1577/

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