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Relatively maximum
volume rigidity in Alexandrov geometry
Nan Li and Xiaochun Rong |
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Vol. 259 (2012), No. 2, 387–420
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 53C21, 53C23, 53C24
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Milestones
Received: 10 December 2011
Accepted: 5 March 2012
Published: 3 October 2012
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Authors |
| Nan Li | |
| Department of Mathematics University of Notre Dame 255 Hurley Hall Notre Dame, IN 46556 United States |
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| 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 |
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