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  • Clément Debin (a1)

We prove a compactness theorem for metrics with bounded integral curvature on a fixed closed surface $\unicode[STIX]{x1D6F4}$ . As a corollary we obtain a new convergence result for sequences of metrics with conical singularities, where an accumulation of singularities is allowed.

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1. Ahlfors, L. V., Conformal Invariants (AMS Chelsea Publishing, Providence, RI, 2010).
2. Ahlfors, L. V., Complex Analysis, third edition (McGraw-Hill Book Co., New York, 1978).
3. Aleksandrov, A. D. and Zalgaller, V. A., Intrinsic Geometry of Surfaces, Translations of Mathematical Monographs, Volume 15 (American Mathematical Society, Providence, RI, 1967). Translated from the Russian by J. M. Danskin.
4. Anderson, M. T., Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102(2) (1990), 429445.
5. Anderson, M. T. and Cheeger, J., C 𝛼 -compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differ. Geom. 35(2) (1992), 265281.
6. Axler, S., Bourdon, P. and Ramey, W., Harmonic Function Theory, Graduate Texts in Mathematics, Volume 137 (Springer, New York, 1992).
7. Bredon, G. E., Topology and Geometry, Graduate Texts in Mathematics, Volume 139 (Springer, New York, 1997).
8. Debin, C., Géométrie des surfaces singulières, PhD thesis, École Doctorale MSTII, Institut Fourier, Grenoble, 2016.
9. DeTurck, D. M. and Kazdan, J. L., Some regularity theorems in Riemannian geometry, Ann. Sci. École Norm. Sup. (4) 14(3) (1981), 249260.
10. Greene, R. E. and Wu, H., Lipschitz convergence of Riemannian manifolds, Pac. J. Math. 131(1) (1988), 119141.
11. Gromov, M., Structures métriques pour les variétés riemanniennes, Textes Mathématiques (Mathematical Texts), Volume 1 (CEDIC, Paris, 1981).
12. Hebey, E. and Herzlich, M., Harmonic coordinates, harmonic radius and convergence of Riemannian manifolds, Rend. Mat. Appl. (7) 17(4) (1997), 569605.
13. Jost, J. and Karcher, H., Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, Manuscripta Math. 40(1) (1982), 2777.
14. Kasue, A., A convergence theorem for Riemannian manifolds and some applications, Nagoya Math. J. 114 (1989), 2151.
15. Peters, S., Convergence of Riemannian manifolds, Compos. Math. 62(1) (1987), 316.
16. Reshetnyak, Y. G., Two-Dimensional Manifolds of Bounded Curvature, Geometry IV, Encyclopaedia Math. Sci. (Springer, Berlin, 1993).
17. Reshetnyak, Y. G., On the conformal representation of Alexandrov surfaces, Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat. (Univ. Jyväskylä, Jyväskylä, 2001).
18. Shioya, T., The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature, Trans. Amer. Math. Soc. 351(5) (1999), 17651801.
19. Troyanov, M., Les surfaces à courbure intégrale bornée au sens d’Alexandrov, Journées annuelles de la SMF (2009), 118.
20. Troyanov, M., Un principe de concentration-compacité pour les suites de surfaces riemanniennes, Ann. Inst. H. Poincaré Anal. Non Linéaire 8(5) (1991), 419441.
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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
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