Finitely bi-Lipschitz homeomorphisms between Finsler manifolds
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Finitely bi-Lipschitz homeomorphisms between Finsler manifolds Elena Afanas’eva1 · Anatoly Golberg2 Received: 4 January 2020 / Revised: 13 August 2020 / Accepted: 4 September 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper we investigate the boundary behavior of finitely bi-Lipschitz homeomorphisms between Finsler manifolds. Our study involves the module technique and classes of mappings whose moduli of the curve/surface families are integrally controlled from above and below. The Lusin (N )-property with respect to the k-dimensional Hausdorff measure for the finitely bi-Lipschitz mappings is also established. Keyword Finsler manifolds, Finitely bi-Lipschitz homeomorphisms, Lower Q-homeomorphisms, Ring Q-homeomorphisms, Moduli of families of surfaces and curves, Boundary behavior of bi-Lipschitz homeomorphisms, Lusin (N )-property Mathematics Subject Classification Primary: 30L10, 58B20; Secondary: 30C65, 53C60
1 Introduction We continue the investigation of the main analytic features of mappings between Finsler manifolds started in [1–3]. The main goal of the work is to establish the connections of finitely bi-Lipschitz homeomorphisms with ring and lower Q-
Dedicated to Professors Samuel Krushkal and Lawrence Zalcman on the occasion of their jubilees
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Anatoly Golberg [email protected] Elena Afanas’eva [email protected]
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Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, 1 Dobrovol’skogo St., Slavyansk 84100, Ukraine
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Department of Mathematics, Holon Institute of Technology, 52 Golomb St., P.O.B. 305, Holon 5810201, Israel 0123456789().: V,-vol
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homeomorphisms on Finsler manifolds. These connections are applied in Sect. 6 in the study of boundary behavior of finitely bi-Lipschitz mappings. We start with the main theorem whose proof will be given in Sect. 5. Theorem 1.1 Let D and D be two domains in Finsler n-dimensional manifolds (M, ) and (M∗ , ∗ ), respectively, n ≥ 2, and let M∗ be a hyperconvex space. If f : D → D is a finitely bi-Lipschitz homeomorphism then f is both lower Q1
homeomorphism with Q = K In−1 (x, f ) and ring Q ∗ -homeomorphism with Q ∗ = 1 stands for the inner dilatation of mapping f , C · K I (x, f ), where K I (x, f ) ∈ L loc and C is a constant arbitrarily close to 1. The definition of hyperconvex spaces is given below in Sect. 4. First we recall some needed definitions related to the theory of Finsler manifolds. Let M be an n-dimensional differentiable manifold, n ≥ 2. By the differentiability we mean C ∞ –differentiability. For a point x ∈ M, Tx M denotes the tangent space at x, and T M := ∪x∈M Tx M is the tangent bundle. The Finsler manifold is a differentiable manifold M equipped the Finsler metric (x, ξ ) : T M → R+ satisfying the conditions: (i) regularity: ∈ C ∞ on T M0 := T M \ {0}; (ii) positive homogeneity: is positive homogeneous that is (x, aξ ) = a(x, ξ ) for all positive a ∈ R and (x, ξ ) > 0 for ξ = 0; 2 2 (x,ξ ) (iii) the Legendre condition or strong convexi
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