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 Tr. Mat. Inst. Steklova, 2011, Volume 273, Pages 192–206 (Mi tm3284)

Filling minimality of Finslerian 2-discs

S. V. Ivanov

St. Petersburg Department of the Steklov Mathematical Institute, Russian Academy of Sciences, St. Petersburg, Russia

Abstract: We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this result for Finsler metrics with area defined as the two-dimensional Holmes–Thompson volume. This implies a generalization of Pu's isosystolic inequality to Finsler metrics, both for the Holmes–Thompson and Busemann definitions of the Finsler area.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2011, 273, 176–190

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UDC: 514.76
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Citation: S. V. Ivanov, “Filling minimality of Finslerian 2-discs”, Modern problems of mathematics, Collected papers. In honor of the 75th anniversary of the Institute, Tr. Mat. Inst. Steklova, 273, MAIK Nauka/Interperiodica, Moscow, 2011, 192–206; Proc. Steklov Inst. Math., 273 (2011), 176–190

Citation in format AMSBIB
\Bibitem{Iva11} \by S.~V.~Ivanov \paper Filling minimality of Finslerian 2-discs \inbook Modern problems of mathematics \bookinfo Collected papers. In honor of the 75th anniversary of the Institute \serial Tr. Mat. Inst. Steklova \yr 2011 \vol 273 \pages 192--206 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3284} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2893545} \zmath{https://zbmath.org/?q=an:1241.53062} \elib{http://elibrary.ru/item.asp?id=16456345} \transl \jour Proc. Steklov Inst. Math. \yr 2011 \vol 273 \pages 176--190 \crossref{https://doi.org/10.1134/S0081543811040079} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000295982500007} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Ivanov S., “Local Monotonicity of Riemannian and Finsler Volume with Respect to Boundary Distances”, Geod. Dedic., 164:1 (2013), 83–96
2. Koehler H., “On Filling Minimality of Simple Finsler Manifolds”, Rev. Mat. Iberoam., 30:1 (2014), 331–348
3. Paiva J.C.A., Balacheff F., Tzanev K., “Isosystolic inequalities for optical hypersurfaces”, Adv. Math., 301 (2016), 934–972
4. Sabourau S., Yassine Z., “Optimal systolic inequalities on Finsler Möbius bands”, J. Topol. Anal., 8:2 (2016), 349–372
5. Lytchak A., Wenger S., “Intrinsic Structure of Minimal Discs in Metric Spaces”, Geom. Topol., 22:1 (2018), 591–644
6. Lytchak A. Wenger S., “Isoperimetric Characterization of Upper Curvature Bounds”, Acta Math., 221:1 (2018), 159–202
7. Gorbachev D.V., Ivanov V.I., Tikhonov S.Yu., “Positive l-P-Bounded Dunkl-Type Generalized Translation Operator and Its Applications”, Constr. Approx., 49:3 (2019), 555–605
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