Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 67, Number 1 (2015), 127-157.
Quasi-isometries and isoperimetric inequalities in planar domains
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.
J. Math. Soc. Japan, Volume 67, Number 1 (2015), 127-157.
First available in Project Euclid: 22 January 2015
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions)
Secondary: 30F20: Classification theory of Riemann surfaces 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 31C12: Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14]
CANTON, Alicia; GRANADOS, Ana; PORTILLA, Ana; RODRIGUEZ, Jose M. Quasi-isometries and isoperimetric inequalities in planar domains. J. Math. Soc. Japan 67 (2015), no. 1, 127--157. doi:10.2969/jmsj/06710127. https://projecteuclid.org/euclid.jmsj/1421936547