# Systolically extremal nonpositively curved surfaces are flat with finitely many singularities

## TL;DR

This paper proves that systolically extremal nonpositively curved surfaces are flat except at finitely many conical singularities, using a decomposition into flat bands and polygonal regions, with bounds on singularities.

## Contribution

It establishes that such extremal surfaces are piecewise flat with finitely many singularities, advancing understanding of their geometric regularity.

## Key findings

- Surfaces are piecewise flat with finitely many conical singularities.
- Decomposition into flat bands and polygonal regions is effective.
- Upper bound of g^{4+ε} on the number of 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 $g^{4+\epsilon}$ on the number of singularities.