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ISRN Geometry
Volume 2011 (2011), Article ID 502814, 7 pages
http://dx.doi.org/10.5402/2011/502814
Research Article

A Restriction for Singularities on Collapsing Orbifolds

Department of Mathematics and Statistics, California State University, Long Beach, CA 90840, USA

Received 8 August 2011; Accepted 5 September 2011

Academic Editors: S. Kar, U. Lindström, and E. H. Saidi

Copyright © 2011 Yu Ding. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Every point 𝑝 in an orbifold 𝑋 has a neighborhood that is homeomorphic to 𝐺 𝑝 ⧡ 𝐡 π‘Ÿ ( 0 ) , where 𝐺 𝑝 is a finite group acting on 𝐡 π‘Ÿ ( 0 ) βŠ‚ ℝ 𝑛 , so that 𝐺 𝑝 ( 0 ) = 0 . Assume 𝑋 is a Riemannian orbifold with isolated singularities that is collapsing, that is, 𝑋 admits a sequence of metrics 𝑔 𝑖 with uniformly bounded curvature, so that, for any π‘₯ ∈ 𝑋 , the volume of 𝐡 1 ( π‘₯ ) , with respect to the metric 𝑔 𝑖 , goes to 0 as 𝑖 β†’ ∞ . For such 𝑋 , we prove that | 𝐺 𝑝 | ≀ ( 2 πœ‹ / 0 . 4 7 ) 𝑛 ( 𝑛 βˆ’ 1 ) for all singularities 𝑝 ∈ 𝑋 .