It is known that the genus two surface admits a piecewise flat metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved metrics. We prove that this piecewise flat metric is also critical for slow metric variations, without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops in different free homotopy classes.

中文翻译：

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收缩期样极值属二面

众所周知，属二曲面允许具有锥形奇点的分段平坦度量，这是所有非正弯曲度量中收缩不等式的极值。我们证明，对于另一种类型的收缩不等式而言，这种分段平坦度量对于涉及不同自由同伦类中最短不可收缩环长度的另一种类型的收缩不等式而言，对于缓慢的度量变化也是至关重要的。