TY - JOUR
T1 - Reciprocal lower bound on modulus of curve families in metric surfaces
AU - Rajala, Kai
AU - Romney, Matthew
N1 - Publisher Copyright:
© 2019 Annales Academiæ Scientiarum Fennicæ.
PY - 2019
Y1 - 2019
N2 - We prove that any metric space X homeomorphic to R2 with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let Q ⊂ X be a topological quadrilateral with boundary edges (in cyclic order) denoted by ζ1, ζ2, ζ3, ζ4 and let γ (ζi, ζj ;Q) denote the family of curves in Q connecting ζi and ζj ; then mod γ (ζ1, ζ3;Q)mod γ (ζ2, ζ4;Q) ≥ 1/κ for κ = 20002 · (4/Π)2. This answers a question in [6] concerning minimal hypotheses under which a metric space admits a quasiconformal parametrization by a domain in R2.
AB - We prove that any metric space X homeomorphic to R2 with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let Q ⊂ X be a topological quadrilateral with boundary edges (in cyclic order) denoted by ζ1, ζ2, ζ3, ζ4 and let γ (ζi, ζj ;Q) denote the family of curves in Q connecting ζi and ζj ; then mod γ (ζ1, ζ3;Q)mod γ (ζ2, ζ4;Q) ≥ 1/κ for κ = 20002 · (4/Π)2. This answers a question in [6] concerning minimal hypotheses under which a metric space admits a quasiconformal parametrization by a domain in R2.
KW - Coarea inequality
KW - Conformal modulus
KW - Quasiconformal mapping
KW - Uniformization
UR - http://www.scopus.com/inward/record.url?scp=85070107472&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85070107472&partnerID=8YFLogxK
U2 - 10.5186/AASFM.2019.4442
DO - 10.5186/AASFM.2019.4442
M3 - Article
AN - SCOPUS:85070107472
SN - 1239-629X
VL - 44
SP - 681
EP - 692
JO - Annales Academiae Scientiarum Fennicae Mathematica
JF - Annales Academiae Scientiarum Fennicae Mathematica
IS - 2
ER -