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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Noguchi, Junjiro / Shahidi, Freydoon / Sogge, Christopher D. / Wienhard, Anna

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Volume 26, Issue 6


On certain optimal diffeomorphisms between closed curves

Andrea Cerri / Barbara Di Fabio
Published Online: 2011-12-10 | DOI: https://doi.org/10.1515/form.2011.172


The concept of natural pseudo-distance has proven to be a powerful tool for measuring the dissimilarity between shape properties of topological spaces, modeled as continuous real-valued functions defined on the spaces themselves. Roughly speaking, the natural pseudo-distance is defined as the infimum of the change of the functions' values, when moving from one space to the other through homeomorphisms, if possible. In this paper, we prove the first available result about the existence of optimal homeomorphisms between closed curves, i.e. inducing a change of the function that equals the natural pseudo-distance. Moreover, we show that, under our assumptions, this optimal homeomorphism is actually a diffeomorphism.

Keywords: Natural pseudo-distance; measuring function; Morse function; size theory

MSC: 57S05; 57S10; 54C30; 68U05

About the article

Received: 2011-05-26

Revised: 2011-09-26

Published Online: 2011-12-10

Published in Print: 2014-11-01

Funding Source: Austrian Science Fund (FWF)

Award identifier / Grant number: P20134-N13

Citation Information: Forum Mathematicum, Volume 26, Issue 6, Pages 1611–1628, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/form.2011.172.

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