Annals of Global Analysis and Geometry

, Volume 37, Issue 3, pp 241–262 | Cite as

Extrinsic curvature of semiconvex subspaces in Alexandrov geometry

  • Stephanie B. Alexander
  • Richard L. BishopEmail author
Original Paper


In Alexandrov spaces of curvature bounded either above (CBA) or below (CBB), we obtain extrinsic curvature bounds on subspaces associated with semiconcave functions. These subspaces play the role in singular geometry of submanifolds in Riemannian geometry, and arise naturally in many different places. For CBA spaces, we obtain new intrinsic curvature bounds on subspaces. For CBB spaces whose boundary is extrinsically curved, we strengthen Perelman’s concavity theorem for distance from the boundary, deriving corollaries on sharp diameter bounds, contractibility, and rigidity.


Alexandrov spaces CAT(K) spaces Semi-concave functions Subspaces Extrinsic curvature 

Mathematics Subject Classification (2000)

53C20 53C70 53B25 58C05 


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.University of Illinois at Urbana-ChampaignUrbanaUSA

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