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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

Volumes and areas of Lipschitz metrics


Author: S. V. Ivanov
Translated by: the author
Original publication: Algebra i Analiz, tom 20 (2008), nomer 3.
Journal: St. Petersburg Math. J. 20 (2009), 381-405
MSC (2000): Primary 53B40
DOI: https://doi.org/10.1090/S1061-0022-09-01053-X
Published electronically: April 7, 2009
MathSciNet review: 2454453
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Abstract: Methods of estimating (Riemannian and Finsler) filling volumes by using nonexpanding maps to Banach spaces of $ L^\infty$-type are developed and generalized. For every Finsler volume functional (such as the Busemann volume or the Holmes-Thompson volume), a natural extension is constructed from the class of Finsler metrics to all Lipschitz metrics, and the notion of area is defined for Lipschitz surfaces in a Banach space. A correspondence is established between minimal fillings and minimal surfaces in $ L^\infty$-type spaces. A Finsler volume functional for which the Riemannian and the Finsler filling volumes are equal is introduced; it is proved that this functional is semielliptic.


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Additional Information

S. V. Ivanov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: svivanov@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-09-01053-X
Keywords: Filling volume, Finsler volume functional, (strong) geodesic minimality property
Received by editor(s): May 29, 2007
Published electronically: April 7, 2009
Additional Notes: Supported by RFBR (grant no. 05-01-00939)
Article copyright: © Copyright 2009 American Mathematical Society