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Geometric analysis demonstrates an optimal systolic area bound in nonpositively curved closed surfaces, establishing minimality for the square flat Klein bottle.
We show that the systolic area of every nonpositively curved closed surface M other than the torus is at least 1, with equality if and only if M is isometric to a square flat Klein bottle. The proof focuses on the Klein-double 4RP² 4 R P 2 and exploits Weil’s isoperimetric inequality and a comparison theorem involving a new kind of exponential-type map.
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Katz et al. (2026) studied this question.
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