Systolically extremal nonpositively curved surfaces are flat with finitely many singularities

Mikhail G. Katz, Stéphane Sabourau

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The regularity of systolically extremal surfaces is a notoriously difficult problem already discussed by Gromov in 1983, who proposed an argument toward the existence of L2-extremizers exploiting the theory of r-regularity developed by White and others by the 1950s. We propose to study the problem of systolically extremal metrics in the context of generalized metrics of nonpositive curvature. A natural approach would be to work in the class of Alexandrov surfaces of finite total curvature, where one can exploit the tools of the completion provided in the context of Radon measures as studied by Reshetnyak and others. However the generalized metrics in this sense still don't have enough regularity. Instead, we develop a more hands-on approach and show that, for each genus, every systolically extremal nonpositively curved surface is piecewise flat with finitely many conical singularities. This result exploits a decomposition of the surface into flat systolic bands and nonsystolic polygonal regions, as well as the combinatorial/topological estimates of Malestein-Rivin-Theran, Przytycki, Aougab-Biringer-Gaster and Greene on the number of curves meeting at most once, combined with a kite excision move. The move merges pairs of conical singularities on a surface of genus g and leads to an asymptotic upper bound g4+ on the number of singularities.

Original languageEnglish
Pages (from-to)319-347
Number of pages29
JournalJournal of Topology and Analysis
Issue number2
StatePublished - Jun 2021

Bibliographical note

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© 2021 World Scientific Publishing Company.


  • Systole
  • extremal metrics
  • nonpositively curved metrics
  • piecewise flat surfaces with conical singularities
  • systolic inequalities


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