TY - JOUR
T1 - Infinitesimally small spheres and conformally invariant metrics
AU - Pouliasis, Stamatis
AU - Solynin, Alexander Yu
N1 - Publisher Copyright:
© 2021, The Hebrew University of Jerusalem.
PY - 2021/6
Y1 - 2021/6
N2 - The modulus metric (also called the capacity metric) on a domain D ⊂ ℝn can be defined as μD(x, y) = inf{cap (D, γ)}, where cap (D, γ) stands for the capacity of the condenser (D, γ) and the infimum is taken over all continua γ ⊂ D containing the points x and y. It was conjectured by J. Ferrand, G. Martin and M. Vuorinen in 1991 that every isometry in the modulus metric is a conformal mapping. In this note, we confirm this conjecture and prove new geometric properties of surfaces that are spheres in the metric space (D, γD).
AB - The modulus metric (also called the capacity metric) on a domain D ⊂ ℝn can be defined as μD(x, y) = inf{cap (D, γ)}, where cap (D, γ) stands for the capacity of the condenser (D, γ) and the infimum is taken over all continua γ ⊂ D containing the points x and y. It was conjectured by J. Ferrand, G. Martin and M. Vuorinen in 1991 that every isometry in the modulus metric is a conformal mapping. In this note, we confirm this conjecture and prove new geometric properties of surfaces that are spheres in the metric space (D, γD).
UR - http://www.scopus.com/inward/record.url?scp=85105426072&partnerID=8YFLogxK
U2 - 10.1007/s11854-021-0152-9
DO - 10.1007/s11854-021-0152-9
M3 - Article
AN - SCOPUS:85105426072
SN - 0021-7670
VL - 143
SP - 179
EP - 205
JO - Journal d'Analyse Mathematique
JF - Journal d'Analyse Mathematique
IS - 1
ER -