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In this note we present a short alternative proof of an estimate obtained byA.B.-Mantel, C.B. Muratov and T.M. Simon in [3] regarding the rigidity ofdegree {$\pm 1$} conformal maps of $\mathbb{S}^2$, i.e. its M\"obiustransformations.

A note on a rigidity estimate for degree $\pm 1$ conformal maps on $\mathbb{S}^2$