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The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature

Author: Takashi Shioya
Journal: Trans. Amer. Math. Soc. 351 (1999), 1765-1801
MSC (1991): Primary 53C20
Published electronically: January 27, 1999
MathSciNet review: 1458311
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Abstract: We study the class of closed $2$-dimensional Riemannian manifolds with uniformly bounded diameter and total absolute curvature. Our first theorem states that this class of manifolds is precompact with respect to the Gromov-Hausdorff distance. Our goal in this paper is to completely characterize the topological structure of all the limit spaces of the class of manifolds, which are, in general, not topological manifolds and even may not be locally $2$-connected. We also study the limit of $2$-manifolds with $L^p$-curvature bound for $p \ge 1$.

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Additional Information

Takashi Shioya
Affiliation: Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan

Keywords: Total curvature, $L^p$-curvature bound, triangle comparison, the Gromov-Hausdorff convergence
Received by editor(s): October 30, 1996
Received by editor(s) in revised form: March 26, 1997
Published electronically: January 27, 1999
Additional Notes: This work was partially supported by a Grant-in-Aid for Scientific Research from the Japan Ministry of Education, Science and Culture.
Article copyright: © Copyright 1999 American Mathematical Society