# Natural pseudodistances between closed surfaces

Pietro Donatini; Patrizio Frosini

Journal of the European Mathematical Society (2007)

- Volume: 009, Issue: 2, page 231-253
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topDonatini, Pietro, and Frosini, Patrizio. "Natural pseudodistances between closed surfaces." Journal of the European Mathematical Society 009.2 (2007): 231-253. <http://eudml.org/doc/277553>.

@article{Donatini2007,

abstract = {Let us consider two closed surfaces $\mathcal \{M\}$, $\mathcal \{N\}$ of class $C^1$ and two functions $\varphi :\mathcal \{M\}\rightarrow \mathbb \{R\}$, $\psi :\mathcal \{N\}\rightarrow \mathbb \{R\}$ of class $C^1$, called measuring functions. The natural pseudodistance $d$ between the pairs $(\mathcal \{M\},)$, $(\mathcal \{N\},\psi )$ is defined as the infimum of $\Theta (f):=\max _\{P\in \mathcal \{M\}\}|\varphi (P)−\psi (f(P))|$ as $f$ varies in the set of all homeomorphisms from $\mathcal \{M\}$ onto $\mathcal \{N\}$. In this paper we prove that the natural pseudodistance equals either $|c_1−c_2|$, $\frac\{1\}\{2\}|c_1−c_2|$, or $\frac\{1\}\{3\}|c_1−c_2|$, where $c_1$ and $c_2$ are two suitable critical values of the measuring functions. This shows that a previous relation between the natural
pseudodistance and critical values obtained in general dimension can be improved in the case of closed surfaces. Our result is based on a theorem by Jost and Schoen concerning harmonic maps
between surfaces.},

author = {Donatini, Pietro, Frosini, Patrizio},

journal = {Journal of the European Mathematical Society},

keywords = {natural pseudodistance; measuring function; harmonic map; natural pseudodistance; measuring function; harmonic map; closed surfaces},

language = {eng},

number = {2},

pages = {231-253},

publisher = {European Mathematical Society Publishing House},

title = {Natural pseudodistances between closed surfaces},

url = {http://eudml.org/doc/277553},

volume = {009},

year = {2007},

}

TY - JOUR

AU - Donatini, Pietro

AU - Frosini, Patrizio

TI - Natural pseudodistances between closed surfaces

JO - Journal of the European Mathematical Society

PY - 2007

PB - European Mathematical Society Publishing House

VL - 009

IS - 2

SP - 231

EP - 253

AB - Let us consider two closed surfaces $\mathcal {M}$, $\mathcal {N}$ of class $C^1$ and two functions $\varphi :\mathcal {M}\rightarrow \mathbb {R}$, $\psi :\mathcal {N}\rightarrow \mathbb {R}$ of class $C^1$, called measuring functions. The natural pseudodistance $d$ between the pairs $(\mathcal {M},)$, $(\mathcal {N},\psi )$ is defined as the infimum of $\Theta (f):=\max _{P\in \mathcal {M}}|\varphi (P)−\psi (f(P))|$ as $f$ varies in the set of all homeomorphisms from $\mathcal {M}$ onto $\mathcal {N}$. In this paper we prove that the natural pseudodistance equals either $|c_1−c_2|$, $\frac{1}{2}|c_1−c_2|$, or $\frac{1}{3}|c_1−c_2|$, where $c_1$ and $c_2$ are two suitable critical values of the measuring functions. This shows that a previous relation between the natural
pseudodistance and critical values obtained in general dimension can be improved in the case of closed surfaces. Our result is based on a theorem by Jost and Schoen concerning harmonic maps
between surfaces.

LA - eng

KW - natural pseudodistance; measuring function; harmonic map; natural pseudodistance; measuring function; harmonic map; closed surfaces

UR - http://eudml.org/doc/277553

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.