Stability of Systolic Inequalities for the Möbius Strip and Klein Bottle

Abstract The systolic area $$\alpha _{sys}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mrow> <mml:mi>sys</mml:mi> </mml:mrow> </mml:msub> </mml:math> of a nonsimply connected compact Riemannian surface ( M , g ) is defined as its area divided by the square of the systole, where the systole is equal to the length of a shortest noncontractible closed curve. The systolic inequality due to Bavard states that on the Klein bottle, the systolic area has the optimal lower bound $$\frac{2\sqrt{2}}{\pi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mrow> <mml:mi>π</mml:mi> </mml:mfrac> </mml:math> . Bavard also constructed metrics of minimal systolic area in any given conformal class. We give an alternative proof of these results, which also yields an estimate on the systolic defect $$\alpha _{sys}-\frac{2\sqrt{2}}{\pi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mrow> <mml:mi>sys</mml:mi> </mml:mrow> </mml:msub> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mrow> <mml:mi>π</mml:mi> </mml:mfrac> </mml:mrow> </mml:math> in terms of the $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -distance of the conformal factor to the metric which minimizes the systolic area. On the Möbius strip, we also prove similar estimates for metrics in fixed conformal classes.