Isometries for the modulus metric in higher dimensions are conformal mappings
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https://doi.org/10.1007/s11425-018-1670-6
Isometries for the modulus metric in higher dimensions are conformal mappings Xiaohui Zhang Department of Mathematical Sciences, Zhejiang Sci-Tech University, Hangzhou 310018, China Email: [email protected] Received October 17, 2018; accepted March 19, 2020
Abstract
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For a proper subdomain D of R and for all x, y ∈ D define µD (x, y) = inf Cap(D, Cxy ), Cxy
where the infimum is taken over all curves Cxy = γ[0, 1] in D with γ(0) = x and γ(1) = y, and Cap denotes the conformal capacity of condensers. The quantity µD is a metric if and only if the domain D has a boundary of positive conformal capacity. If Cap(∂D) > 0, we call µD the modulus metric of D. Ferrand et al. (1991) have conjectured that isometries for the modulus metric are conformal mappings. Very recently, this conjecture has been proved for n = 2 by Betsakos and Pouliasis (2019). In this paper, we prove that the conjecture is also true in higher dimensions n > 3. Keywords MSC(2010)
capacity, modulus metric, isometry, M¨ obius transformation 30C65
Citation: Zhang X H. Isometries for the modulus metric in higher dimensions are conformal mappings. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-018-1670-6
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Introduction n
Let D ⊂ R be an open set and let K ⊂ D be a compact set. The pair (D, K) is called a condenser. If D ⊂ Rn and p > 1, the p-capacity of the condenser (D, K) is defined by ∫ p-Cap(D, K) = inf |∇u|p dmn , u
Rn
where mn stands for the n-dimensional Lebesgue measure, and the infimum is taken over the family of all the non-negative ACLp functions u with compact support in D such that u(x) > 1 for x ∈ K. Here ( ) ∂u ∂u ∇u(x) = (x), . . . , (x) . ∂x1 ∂xn A function is called ACLp if it is absolutely continuous on lines and its partial derivatives are locally Lp integrable. If p = n we denote n-Cap(D, K) simply by Cap(D, K) and call it the capacity or conformal n capacity of the condenser (D, K). The conformal capacity is conformally invariant. For D ⊂ R , if c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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∞ ∈ D, the conformal capacity is defined by use of a M¨obius transformation to avoid the infinity. The notion of conformal capacity was introduced by Loewner [25] and has been extensively developed for Rn (see, for example, [11, 12]). The general p-capacity has been studied in partial differential equations and nonlinear potential theory (see, for example, [20, 22, 26]). A compact set K ⊂ Rn is said to be of capacity zero, denoted by Cap(K) = 0, if there exists a bounded n open set D with K ⊂ D and Cap(D, K) = 0. A compact set K ( R , is said to be of capacity zero if K can be mapped by a M¨obius transformation onto a bounded set of capacity zero. Otherwise, K is said to be of positive capacity, and we write Cap(K) > 0. n For a proper subdomain D of R and for all x, y ∈ D, define µD (x, y) = inf Cap(D, Cxy ), Cxy
where the infimum is taken over all curves Cxy =
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