This paper studies the stability of isoperimetric inequalities under quasi-isometries between non-exceptional Riemann surfaces endowed with their Poincaré metrics. This stability was proved by Kanai in the more general setting of Riemannian manifolds under the condition of positive injectivity radius. The present work proves the stability of the linear isoperimetric inequality for planar surfaces (genus zero surfaces) without any condition on their injectivity radii. It is also shown the stability of any non-linear isoperimetric inequality.
"Quasi-isometries and isoperimetric inequalities in planar domains." J. Math. Soc. Japan 67 (1) 127 - 157, January, 2015. https://doi.org/10.2969/jmsj/06710127