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Topological Properties of Regular Boundary Domains in Riemannian Manifolds

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Abstract

Various topological properties of the smooth, Lipschitz, convex, quasiconvex, uniform and QED domains in Riemannian n-dimensional manifolds are investigated. It has been established that Riemannian manifolds are weakly flat spaces in the terminology of general metric measure spaces. Moreover, several sufficient conditions on continuous and homeomorphic extensions to the boundary of \(\eta \)-quasisymmetries, weakly H-quasisymmetries and \(\omega \)-quasimöbius homeomorphisms between domains in Riemannian manifolds are obtained. The relationship between the above mapping classes is provided too.

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References

  1. Afanas’eva, E.S.: Boundary behavior of ring \(Q\)-homeomorphisms on Riemannian manifolds. Ukr. Math. J. 63(10), 1479–1493 (2012)

    Article  MathSciNet  Google Scholar 

  2. Afanas’eva, E.S.: On the boundary behavior of one class of mappings in metric spaces. Ukr. Math. J. 66(1), 16–29 (2014)

    Article  MathSciNet  Google Scholar 

  3. Afanas’eva, E., Golberg, A.: Homeomorphisms of finite metric distortion between Riemannian manifolds. Comput. Methods Funct. Theory (2022). https://doi.org/10.1007/s40315-021-00431-3

    Article  MATH  Google Scholar 

  4. Afanas’eva, E., Golberg, A.: Finitely bi-Lipschitz homeomorphisms between Finsler manifolds. Anal. Math. Phys. 10(4), 1–16 (2020)

    Article  MathSciNet  Google Scholar 

  5. Aseev, V.V.: The quasimöbius property on small circles and quasiconformality. Sib. Math. J. 54(2), 196–204 (2013)

    Article  MathSciNet  Google Scholar 

  6. Beurling, A., Ahlfors, L.: The boundary correspondence under quasiconformal mappings. Acta Math. 96, 125–142 (1956)

    Article  MathSciNet  Google Scholar 

  7. Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Pure and Applied Mathematics, vol. 120, 2nd edn. Academic Press, Cambridge (1986)

    MATH  Google Scholar 

  8. Bourbaki, N.: General Topology: Chapters 1–4 (1989)

  9. Brickell, F., Clark, R.S.: Differentiable Manifolds. Van Nostrand Reinhold, London (1970)

    MATH  Google Scholar 

  10. Gehring, F.W., Martio, O.: Quasiextremal distance domains and extension of quasiconformal mappings. J. Anal. Math. 45, 181–206 (1985)

    Article  MathSciNet  Google Scholar 

  11. Golberg, A.: Geometric characteristics of mappings with first generalized derivatives. Rev. Roumaine Math. Pures Appl. 47(5–6), 673–682 (2002–2003)

  12. Golberg, A.: Generalized classes of quasiconformal homeomorphisms. Math. Rep. (Bucur.) 7(57), 289–303 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Hakobyan, H., Herron, A.: Euclidean quasiconvexity. Ann. Acad. Sci. Fenn. Ser. A1 Math. 33, 205–230 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)

    Book  Google Scholar 

  15. Heinonen, J., Koskela, P.: Quasiconformal maps on metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998)

    Article  MathSciNet  Google Scholar 

  16. Jost, J.: Riemannian Geometry and Geometric Analysis, 3rd edn. Springer, Berlin (2002)

    Book  Google Scholar 

  17. Koskela, P., Wildrick, K.: Analytic Properties of Quasiconformal Mappings Between Metric Spaces. Metric and Differential Geometry. Progress in Mathematics, vol. 297, pp. 163–174. Springer, Basel (2012)

    MATH  Google Scholar 

  18. Kovtonyuk,D.A., Ryazanov,V.I., Salimov, R.R., Sevost’yanov, E.A.: Boundary behavior of Orlicz–Sobolev classes. Translation of Mat. Zametki 95(4), 564–576 (2014). Math. Notes 95(3–4), 509–519 (2014)

  19. Kuratowski, K.: Topology, vol. II. Academic Press, New York (1968)

    MATH  Google Scholar 

  20. Lee, J.M.: Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics, vol. 176. Springer, New York (1997)

    Book  Google Scholar 

  21. Martio, O.: Definitions for uniform domains. Ann. Acad. Sci. Fenn. Ser. A1 Math. 5, 197–205 (1980)

    Article  MathSciNet  Google Scholar 

  22. Martio, O., Ryazanov, V., Srebro, U., Yakubov, E.: Moduli in Modern Mapping Theory. Springer Monographs in Mathematics. Springer, New York (2009)

    MATH  Google Scholar 

  23. Martio, O., Sarvas, J.: Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A1. Math. 4, 383–401 (1978/1979)

  24. Stone, A.H.: Paracompactness and product spaces. Bull. Am. Math. Soc. 54, 977–982 (1948)

    Article  MathSciNet  Google Scholar 

  25. Suominen, K.: Quasiconformal maps in manifolds. Ann. Acad. Sci. Fenn. Ser. A1 Math. 393, 1–39 (1966)

    MathSciNet  MATH  Google Scholar 

  26. Tukia, P., Väisälä, J.: Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. AI Math. 5, 97–114 (1980)

    Article  MathSciNet  Google Scholar 

  27. Tyson, J.: Quasiconformality and quasisymmetry in metric measure spaces. Ann. Acad. Sci. Fen. Math. 23, 525–548 (1998)

    MathSciNet  MATH  Google Scholar 

  28. Väisälä, J.: Lectures on n-Dimensional Quasiconformal Mappings. Lecture Notes in Mathematics, vol. 229. Springer, Berlin (1971)

    Book  Google Scholar 

  29. Väisälä, J.: Quasimöbius maps. J. d’Anal. Math. 44, 218–234 (1984/1985)

  30. Väisälä, J.: Invariants for quasisymmetric, quasi-Möbius and bi-Lipschitz mappings. J. Anal. Math. 50, 201–223 (1988)

    Article  Google Scholar 

  31. Väisälä, J.: Uniform domains. Tohoku Math. J. 40, 101–118 (1988)

    Article  MathSciNet  Google Scholar 

  32. Väisälä, J.: Domains and Maps. Lectures Notes in Mathematics, vol. 1508, pp. 119–131. Springer, Berlin (1992)

  33. Väisälä, J., Vuorinen, M., Wallin, H.: Thick sets and quasisymmetric maps. Nagoya Math. J. 135, 121–148 (1994)

    Article  MathSciNet  Google Scholar 

  34. Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, vol. 94. Springer, New York (1983)

  35. Whyburn, G.T.: Analytic Topology. American Mathematics Society, Providence (1942)

    Book  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, 1 Dobrovol’skogo St., Slavyansk, 84100, Ukraine

    Elena Afanas’eva

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  1. Elena Afanas’eva
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Communicated by David Shoikhet.

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This article is part of the topical collection “Higher Dimensional Geometric Function Theory and Hypercomplex Analysis” edited by Irene Sabadini, Michael Shapiro and Daniele Struppa.

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Afanas’eva, E. Topological Properties of Regular Boundary Domains in Riemannian Manifolds. Complex Anal. Oper. Theory 16, 39 (2022). https://doi.org/10.1007/s11785-022-01212-z

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