Natural pseudodistances between closed surfaces

Donatini, 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 -