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+𝜖 on the number of singularities.

中文翻译：

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收缩极值非正曲面是平坦的，具有有限多个奇点

收缩极值曲面的规律性是一个众所周知的难题，Gromov 在 1983 年已经讨论过，他提出了一个关于存在大号2-利用理论的极端分子r- White 和其他人在 1950 年代开发的规律性。我们建议在非正曲率的广义度量的背景下研究收缩极值度量的问题。一种自然的方法是在有限总曲率的 Alexandrov 曲面类中工作，其中可以利用 Reshetnyak 和其他人研究的氡测量上下文中提供的完成工具。然而，这种意义上的广义度量仍然没有足够的规律性。相反，我们开发了一种更实际的方法，并表明，对于每个属，每个收缩极值非正曲面都是分段平坦的，具有有限多个锥形奇点。该结果利用将表面分解为平坦收缩带和非收缩多边形区域，以及 Malestein-Rivin-Theran、Przytycki、Aougab-Biringer-Gaster 和 Greene 对最多一次相遇的曲线数量的组合/拓扑估计，并结合风筝切除移动。该移动合并了属表面上的锥形奇点对G并导致渐近上界G4+𝜖关于奇点的数量。