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no. 3
Metrics and convergence in moduli spaces of maps
Joseph Palmer1
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, Serie 6, Volume 27 (2018) no. 3, pp. 497-526.

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 ⊂M,φ:B φ →NwithB φ ,φbothmeasurable} 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.

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 ⊂M,φ:B φ →NavecB φ ,φmesurable} 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.

Received:
Accepted:
Published online:
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.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Joseph Palmer. Metrics and convergence in moduli spaces of maps. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 3, pp. 497-526. doi : 10.5802/afst.1577. https://afst.centre-mersenne.org/articles/10.5802/afst.1577/

[1] Miguel Abreu Topology of symplectomorphism groups of S 2 ×S 2 , Invent. Math., Volume 131 (1998) no. 1, pp. 1-24 | DOI | MR | Zbl

[2] Paul Biran Connectedness of spaces of symplectic embeddings, Int. Math. Res. Not., Volume 1996 (1996) no. 10, pp. 487-491 | DOI | MR | Zbl

[3] Alexander Brudnyi; Yuri Brudnyi Methods of geometric analysis in extension and trace problems. Vol. 1, Monographs in Mathematics, 102, Birkhäuser, 2012, xxiii+560 pages | Zbl

[4] Gerald B. Folland Real analysis. Modern techniques and their applications, Pure and Applied Mathematics, Wiley, 1999, xiv+386 pages | Zbl

[5] Mikhael L. Gromov Pseudoholomorphic curves in symplectic manifolds, Invent. Math., Volume 82 (1985), pp. 307-347 | DOI | Zbl

[6] Heinrich H. W. Hopf; Willi Rinow Über den Begriff der vollständigen differentialgeometrischen Fläche, Comment. Math. Helv., Volume 3 (1931) no. 1, pp. 209-225 | DOI | Zbl

[7] John M. Lee Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 2018, Springer, 2006, xvii+628 pages | Zbl

[8] John D. McCarthy; Jon G. Wolfson Symplectic gluing along hypersurfaces and resolution of isolated orbifold singularities, Invent. Math., Volume 119 (1995) no. 1, pp. 129-154 | DOI | MR | Zbl

[9] Jürgen Moser On the volume elements on a manifold, Trans. Am. Math. Soc., Volume 120 (1965), pp. 286-294 | DOI | MR | Zbl

[10] Katsumi Nomizu; Hideki Ozeki The existence of complete Riemannian metrics, Proc. Am. Math. Soc., Volume 12 (1961) no. 6, pp. 889-891 | DOI | MR | Zbl

[11] Álvaro Pelayo; Ngọc, San Vũ Hofer’s question on intermediate symplectic capacities, Proc. Lond. Math. Soc., Volume 110 (2015) no. 4, pp. 787-804 | DOI | MR | Zbl

[12] Álvaro Pelayo; Ngọc, San Vũ Sharp symplectic embeddings of cylinders, Indag. Math., Volume 27 (2016) no. 1, pp. 307-317 | DOI | MR | Zbl

[13] Álvaro Pelayo Topology of spaces of equivariant symplectic embeddings, Proc. Am. Math. Soc., Volume 135 (2007) no. 1, pp. 277-288 | DOI | MR | Zbl

[14] Álvaro Pelayo; Ana Rita Pires; Tudor S. Ratiu; Silvia Sabatini Moduli spaces of toric manifolds, Geom. Dedicata, Volume 169 (2014), pp. 323-341 | DOI | MR | Zbl

[15] Ana Cannas da Silva Lectures on Symplectic Geometry, Lecture Notes in Math., 1764, Springer, 2008 | MR | Zbl

[16] Hassler Whitney Analytic extensions of differentiable functions defined in closed sets, Trans. Am. Math. Soc., Volume 36 (1934) no. 1, pp. 63-89 | DOI | MR | Zbl

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