Vol. 259, No. 2, 2012

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Relatively maximum volume rigidity in Alexandrov geometry

Nan Li and Xiaochun Rong

Vol. 259 (2012), No. 2, 387–420
Abstract

Given a compact metric space Z with Hausdorff dimension n, if X is a metric space such that there exists a distance-nonincreasing onto map f : Z X, then the Hausdorff n-volumes satisfy vol(X) vol(Z). The relatively maximum volume conjecture says that if X and Z are both Alexandrov spaces and vol(X) = vol(Z), X is isometric to a gluing space produced from Z along its boundary ∂Z and f is length-preserving. We partially verify this conjecture and give a further classification for compact Alexandrov n-spaces with relatively maximum volume in terms of a fixed radius and space of directions. We also give an elementary proof for a pointed version of the Bishop–Gromov relative volume comparison with rigidity in Alexandrov geometry.

Keywords
volume, radius, Alexandrov space, rigidity, stability
Mathematical Subject Classification 2010
Primary: 53C21, 53C23, 53C24
Milestones
Received: 10 December 2011
Accepted: 5 March 2012
Published: 3 October 2012
Authors
Nan Li
Department of Mathematics
University of Notre Dame
255 Hurley Hall
Notre Dame, IN 46556
United States
Xiaochun Rong
Department of Mathematics
Capital Normal University
105 Xi San Huan Bei Rd.
Haidian District
Beijing 100048
China
Department of Mathematics
Rutgers University
Hill Center for the Mathematical Sciences
110 Frelinghuysen Rd
Piscataway, New Jersey 08854
United States