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A COMPACTNESS THEOREM FOR SURFACES WITH BOUNDED INTEGRAL CURVATURE

Part of: Global differential geometry

Published online by Cambridge University Press:  10 April 2018

Clément Debin
Clément Debin*
Affiliation:
Institut Fourier, Mathematics, Saint Martin d’Heres, France, 38402 ([email protected])
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Abstract

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.

Keywords

differential geometrymetric geometryconical singularitiesAlexandrov surfaces with bounded integral curvaturecompactness theorem

MSC classification

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 2 , March 2020 , pp. 597 - 645
Copyright
© Cambridge University Press 2018

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Footnotes

This research is supported by the ERC advanced grant 320939, Geometry and Topology of Open Manifolds (GETOM).

References

Ahlfors, L. V., Conformal Invariants (AMS Chelsea Publishing, Providence, RI, 2010).Google Scholar
Ahlfors, L. V., Complex Analysis, third edition (McGraw-Hill Book Co., New York, 1978).Google Scholar
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.Google Scholar
Anderson, M. T., Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102(2) (1990), 429–445.Google Scholar
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), 265–281.Google Scholar
Axler, S., Bourdon, P. and Ramey, W., Harmonic Function Theory, Graduate Texts in Mathematics, Volume 137 (Springer, New York, 1992).Google Scholar
Bredon, G. E., Topology and Geometry, Graduate Texts in Mathematics, Volume 139 (Springer, New York, 1997).Google Scholar
Debin, C., Géométrie des surfaces singulières, PhD thesis, École Doctorale MSTII, Institut Fourier, Grenoble, 2016.Google Scholar
DeTurck, D. M. and Kazdan, J. L., Some regularity theorems in Riemannian geometry, Ann. Sci. École Norm. Sup. (4) 14(3) (1981), 249–260.Google Scholar
Greene, R. E. and Wu, H., Lipschitz convergence of Riemannian manifolds, Pac. J. Math. 131(1) (1988), 119–141.Google Scholar
Gromov, M., Structures métriques pour les variétés riemanniennes, Textes Mathématiques (Mathematical Texts), Volume 1 (CEDIC, Paris, 1981).Google Scholar
Hebey, E. and Herzlich, M., Harmonic coordinates, harmonic radius and convergence of Riemannian manifolds, Rend. Mat. Appl. (7) 17(4) (1997), 569–605.Google Scholar
Jost, J. and Karcher, H., Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, Manuscripta Math. 40(1) (1982), 27–77.Google Scholar
Kasue, A., A convergence theorem for Riemannian manifolds and some applications, Nagoya Math. J. 114 (1989), 21–51.Google Scholar
Peters, S., Convergence of Riemannian manifolds, Compos. Math. 62(1) (1987), 3–16.Google Scholar
Reshetnyak, Y. G., Two-Dimensional Manifolds of Bounded Curvature, Geometry IV, Encyclopaedia Math. Sci. (Springer, Berlin, 1993).Google Scholar
Reshetnyak, Y. G., On the conformal representation of Alexandrov surfaces, Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat. (Univ. Jyväskylä, Jyväskylä, 2001).Google Scholar
Shioya, T., The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature, Trans. Amer. Math. Soc. 351(5) (1999), 1765–1801.Google Scholar
Troyanov, M., Les surfaces à courbure intégrale bornée au sens d’Alexandrov, Journées annuelles de la SMF (2009), 1–18.Google Scholar
Troyanov, M., Un principe de concentration-compacité pour les suites de surfaces riemanniennes, Ann. Inst. H. Poincaré Anal. Non Linéaire 8(5) (1991), 419–441.Google Scholar