Abstract
Consider a domain in Euclidean space whose volume element is induced by some weight function, while the arclength element of a curve at a point depends not only on the point, but also on the direction of motion along the curve. In this case we say that an abstract surface is defined over this domain. We prove a version of symmetry principle for the modulus of a family of curves on an abstract surface. In the weighted case we establish that the modulus is continuous when the arclength element is given in the isothermal coordinates.
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Notes
The terms of an at most countable collection \( \{\Gamma_{i}\}_{i\in I} \) of families of curves are separated whenever there exists a tuple \( \{E_{i}\}_{i\in I} \) of disjoint Borel sets such that \( \int\nolimits_{\gamma}\chi_{{}^{n}\setminus E_{i}}\,ds=0 \) for every locally rectifiable curve \( \gamma\in\Gamma_{i} \), whatever \( i\in I \).
Henceforth \( e_{n} \) stands for the vector in \( {}^{n} \) with entries \( (0,\dots,0,1) \).
By the Rademacher Theorem, \( \varphi \) is differentiable almost everywhere in \( {}^{n} \).
Of course, the substantial case is \( J\cap(a^{\prime},b^{\prime})\neq\varnothing \).
References
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Funding
The author was supported by the Mathematical Center in Akademgorodok under Agreement 075–15–2022–281 on April 5, 2022 with the Ministry of Science and Higher Education of the Russian Federation.
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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 3, pp. 659–671. https://doi.org/10.33048/smzh.2022.63.314
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Tryamkin, M.V. Some Properties of the Modulus of a Family of Curves on an Abstract Surface. Sib Math J 63, 548–558 (2022). https://doi.org/10.1134/S0037446622030144
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DOI: https://doi.org/10.1134/S0037446622030144