Journal of the Mathematical Society of Japan

Quasi-isometries and isoperimetric inequalities in planar domains

Alicia CANTON, Ana GRANADOS, Ana PORTILLA, and Jose M. RODRIGUEZ

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Abstract

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.

Article information

Source
J. Math. Soc. Japan, Volume 67, Number 1 (2015), 127-157.

Dates
First available in Project Euclid: 22 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1421936547

Digital Object Identifier
doi:10.2969/jmsj/06710127

Mathematical Reviews number (MathSciNet)
MR3304016

Zentralblatt MATH identifier
1311.30016

Subjects
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]

Keywords
Riemann surface Poincaré metric isoperimetric inequality linear isoperimetric inequality quasi-isometry

Citation

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


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