Annales Academię Scientiarum Fennicę

Mathematica

Volumen 35, 2010, 3-21

# A MODULUS FOR CURVES FROM DISTANCE

## Neil N. Katz

New York City College of Technology, Mathematics Department

300 Jay Street, Brooklyn, NY 11201, U.S.A.; nkatz 'at' citytech.cuny.edu

**Abstract.**
In this paper a modulus of curves
is defined using pseudo-distance functions.
This leads to a notion of quasiconformal maps that is equivalent to the
standard definition when the distance function is Riemannian. The moduli of
families of curves whose endpoints lie in the boundary of open subsets of a
compact, convex set are determined. This allows bounds on volumes of images
of Euclidean balls under quasiconformal maps to be made. Also, certain
generalized, conformal, isosystolic constants are found. Estimates
are given of how these constants, and of how norms of weak upper gradients,
vary under quasiconformal maps.

**2000 Mathematics Subject Classification:**
Primary 30C65, 53C23, 53C65.

**Key words:**
Quasiconformal maps, volume distortion,
conformal isosystolic inequalities, conformal filling volume.

**Reference to this article:** N.N. Katz:
A modulus for curves from distance.
Ann. Acad. Sci. Fenn. Math. 35 (2010), 3-21.

Full document as PDF file

doi:10.5186/aasfm.2010.3501

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