Systolically extremal nonpositively curved surfaces are flat with finitely many singularities

Abstract

The regularity of systolically extremal surfaces is a notoriously difficult problem already discussed by M. Gromov in 1983, who proposed an argument toward the existence of L 2 -extremizers exploiting the theory of r-regularity developed by P. A. 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 g 4+ǫ on the number of singularities.

Sharp reverse isoperimetric inequalities in nonpositively curved cones

“…We would like to provide some context for our study of reverse isoperimetric inequalities. We found the geometric inequalities of this paper while working on a proof that systolically extremal nonpositively curved surfaces are flat with finitely many conical singularities; see [7]. In that context, it was clear that it should be imposssible to round off a conical singularity of angle greater than 2π in a nonsystolic region in order to decrease the area (while keeping the nonpositive curvature condition) by any cut-and-paste argument with metric disks of the same perimeter.…”

Sharp upper bounds on the length of the shortest closed geodesic on complete punctured spheres of finite area

“…In both cases, the extremal nonpositively curved metrics are piecewise flat with conical singularities. It was later proved that this structure is common to all extremal nonpositively curved surfaces; see [26]. Extremal systolic inequalities in a fixed conformal class have been investigated in relation with closed string field theory; see [18,19,30] for the most recent contributions and the references therein.…”

Extremal isosystolic metrics with multiple bands of crossing geodesics

“…The conformal specialization has also been explored in [10]. The case of metrics with non-positive curvature has been studied in [11,12].…”