##plugins.themes.bootstrap3.article.main##
Abstract
In an attempt to develop higher-dimensional quasiconformal mappings on metric measure spaces with curvature conditions, i.e. from Ahlfors to Alexandrov, we show that for n≥2 a noncollapsed RCD(0,n) space with Euclidean volume growth is an n-Loewner space and satisfies the infinitesimal-to-global principle.
Keywords:
quasiconformal mapping, quasi-symmetric, RCD spaces, infinitesimal-to-global principle
##plugins.themes.bootstrap3.article.details##
How to Cite
Deng, J. (2023). Quasiconformal mappings and curvatures on metric measure spaces. Proceedings of the International Geometry Center, 15(3-4), 203-218. https://doi.org/10.15673/tmgc.v15i3-4.2369
Issue
Section
Papers
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution (CC-BY) 4.0 License that allows others to share the work with an acknowledgment of the work’s authorship and initial publication in this journal.
Provided they are the owners of the copyright to their work, authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal’s published version of the work (e.g., post it to an institutional repository, in a journal or publish it in a book), with an acknowledgment of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories, disciplinary repositories, or on their website) prior to and during the submission process.
References
1. L. V. Ahlfors. An extension of Schwarz's lemma. Trans. Amer. Math. Soc., 43(3):359-364, 1938. https://doi.org/10.2307/1990065 pathdoi:10.2307/1990065.
2. L. V. Ahlfors. Lectures on quasiconformal mappings, volume 38 of University Lecture Series. American Mathematical Society, Providence, RI, second edition, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. https://doi.org/10.1090/ulect/038 pathdoi:10.1090/ulect/038.
3. V. Alberge and A. Papadopoulos. A commentary on Lavrentieff's paper \"Sur une classe de representations continues\". In Handbook of Teichmuller theory. Vol. VII, volume 30 of IRMA Lect. Math. Theor. Phys., pages 441-451. Eur. Math. Soc., Zurich, 2020.
4. V. Alberge and A. Papadopoulos. On five papers by Herbert Grotzsch. In Handbook of Teichmuller theory. Vol. VII, volume 30 of IRMA Lect. Math. Theor. Phys., pages 393-415. Eur. Math. Soc., Zurich, 2020.
5. A. D. Aleksandrov. Vnutrennyaya Geometriya Vypuklyh Poverhnosteui. OGIZ, Moscow-Leningrad, 1948.
6. L. Ambrosio. Calculus, heat flow and curvature-dimension bounds in metric measure spaces. In Proceedings of the International Congress of Mathematicians-Rio de Janeiro 2018. Vol. I. Plenary lectures, pages 301-340. World Sci. Publ., Hackensack, NJ, 2018.
7. L. Ambrosio, S. Di Marino, and N. Gigli. Perimeter as relaxed Minkowski content in metric measure spaces. Nonlinear Anal., 153:78-88, 2017. https://doi.org/10.1016/j.na.2016.03.010 pathdoi:10.1016/j.na.2016.03.010.
8. L. Ambrosio, A. Mondino, and G. Savare. Nonlinear diffusion equations and curvature conditions in metric measure spaces. Mem. Amer. Math. Soc., 262(1270):v+121, 2019. https://doi.org/10.1090/memo/1270 pathdoi:10.1090/memo/1270.
9. Z. M. Balogh and A. Kristaly. Sharp isoperimetric and sobolev inequalities in spaces with nonnegative ricci curvature. Mathematische Annalen, March 2022. https://doi.org/10.1007/s00208-022-02380-1 pathdoi:10.1007/s00208-022-02380-1.
10. D. Burago, Y. Burago, and S. Ivanov. A course in metric geometry, volume 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. https://doi.org/10.1090/gsm/033 pathdoi:10.1090/gsm/033.
11. G. Carron. Euclidean volume growth for complete Riemannian manifolds. Milan J. Math., 88(2):455-478, 2020. https://doi.org/10.1007/s00032-020-00321-8 pathdoi:10.1007/s00032-020-00321-8.
12. F. Cavalletti and E. Milman. The globalization theorem for the curvature-dimension condition. Invent. Math., 226(1):1-137, 2021. https://doi.org/10.1007/s00222-021-01040-6 pathdoi:10.1007/s00222-021-01040-6.
13. J. Cheeger and T. H. Colding. On the structure of spaces with Ricci curvature bounded below. I. J. Differential Geom., 46(3):406-480, 1997. URL: http://projecteuclid.org/euclid.jdg/1214459974.
14. J. Deng. Curvature-dimension condition meets Gromov's n-volumic scalar curvature. SIGMA Symmetry Integrability Geom. Methods Appl., 17:Paper No. 013, 20, 2021. https://doi.org/10.3842/SIGMA.2021.013 pathdoi:10.3842/SIGMA.2021.013.
15. J. Deng. Enlargeable length-structure and scalar curvatures. Ann. Global Anal. Geom., 60(2):217-230, 2021. https://doi.org/10.1007/s10455-021-09772-7 pathdoi:10.1007/s10455-021-09772-7.
16. J. Deng. Foliated Positive Scalar Curvature. PhD thesis, University of Goettingen, 2021. https://doi.org/10.53846/goediss-8930 pathdoi:10.53846/goediss-8930.
17. Q. Deng. Holder continuity of tangent cones in RCD(K,N) spaces and applications to non-branching. arXiv e-prints, page arXiv:2009.07956, September 2020. http://arxiv.org/abs/2009.07956 patharXiv:2009.07956.
18. M. Erbar, K. Kuwada, and K.-T. Sturm. On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces. Invent. Math., 201(3):993-1071, 2015. https://doi.org/10.1007/s00222-014-0563-7 pathdoi:10.1007/s00222-014-0563-7.
19. R. Fehlmann and M. Vuorinen. Mori's theorem for n-dimensional quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math., 13(1):111-124, 1988. https://doi.org/10.5186/aasfm.1988.1304 pathdoi:10.5186/aasfm.1988.1304.
20. F. P. Gardiner and N. Lakic. Quasiconformal Teichmuller theory, volume 76 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000. https://doi.org/10.1090/surv/076 pathdoi:10.1090/surv/076.
21. F. W. Gehring, G. J. Martin, and B. P. Palka. An introduction to the theory of higher-dimensional quasiconformal mappings, volume 216 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2017. https://doi.org/10.1090/surv/216 pathdoi:10.1090/surv/216.
22. F. W. Gehring and J. Vaisala. Hausdorff dimension and quasiconformal mappings. J. Lond. Math. Soc. (2), 6:504-512, 1973. https://doi.org/10.1112/jlms/s2-6.3.504 pathdoi:10.1112/jlms/s2-6.3.504.
23. N. Gigli. Nonsmooth differential geometry - an approach tailored for spaces with Ricci curvature bounded from below. Mem. Amer. Math. Soc., 251(1196):v+161, 2018. https://doi.org/10.1090/memo/1196 pathdoi:10.1090/memo/1196.
24. N. Gigli and E. Pasqualetto. Lectures on nonsmooth differential geometry, volume 2 of SISSA Springer Series. Springer, Cham, 2020. https://doi.org/10.1007/978-3-030-38613-9 pathdoi:10.1007/978-3-030-38613-9.
25. H. Grotzsch. Uber die Verzerrung bei schlichten nichtkonformen Abbildungen und uber eine damit zusammenhangende Erweiterung des Picardschen Satzes., 1928.
26. H. Grotzsch. Uber moglichst konforme Abbildungen von schlichten Bereichen., 1932.
27. D. X. Gu, W. Zeng, L. M. Lui, F. Luo, and S.-T. Yau. Recent development of computational conformal geometry. In Fifth International Congress of Chinese Mathematicians. Part 1, 2, volume 2 of AMS/IP Stud. Adv. Math., 51, pt. 1, pages 515-560. Amer. Math. Soc., Providence, RI, 2012.
28. J. Heinonen and P. Koskela. Quasiconformal maps in metric spaces with controlled geometry. Acta Math., 181(1):1-61, 1998. https://doi.org/10.1007/BF02392747 pathdoi:10.1007/BF02392747.
29. S. Honda. New differential operator and noncollapsed RCD spaces. Geom. Topol., 24(4):2127-2148, 2020. https://doi.org/10.2140/gt.2020.24.2127 pathdoi:10.2140/gt.2020.24.2127.
30. H. Huang and X.-T. Huang. Almost splitting maps, transformation theorems and smooth fibration theorems. arXiv e-prints, page arXiv:2207.10029, July 2022. http://arxiv.org/abs/2207.10029 patharXiv:2207.10029.
31. V. Kapovitch and A. Mondino. On the topology and the boundary of N-dimensional mathrmRCD(K,N) spaces. Geom. Topol., 25(1):445-495, 2021. https://doi.org/10.2140/gt.2021.25.445 pathdoi:10.2140/gt.2021.25.445.
32. S. Keith and X. Zhong. The Poincare inequality is an open ended condition. Ann. of Math. (2), 167(2):575-599, 2008. https://doi.org/10.4007/annals.2008.167.575 pathdoi:10.4007/annals.2008.167.575.
33. M. Lavrentieff. On a class of continuous representations. In Handbook of Teichmuller theory. Volume VII, pages 417-439. Berlin: European Mathematical Society (EMS), 2020. https://doi.org/10.4171/203-1/20 pathdoi:10.4171/203-1/20.
34. J. Lott and C. Villani. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2), 169(3):903-991, 2009. https://doi.org/10.4007/annals.2009.169.903 pathdoi:10.4007/annals.2009.169.903.
35. L. M. Lui, T. W. Wong, W. Zeng, X. Gu, P. M. Thompson, T. F. Chan, and S.-T. Yau. A survey on recent development in computational quasi-conformal geometry and its applications. In Fifth International Congress of Chinese Mathematicians. Part 1, 2, volume 2 of AMS/IP Stud. Adv. Math., 51, pt. 1, pages 697-717. Amer. Math. Soc., Providence, RI, 2012. https://doi.org/10.3934/ipi.2010.4.311 pathdoi:10.3934/ipi.2010.4.311.
36. R. Mane, P. Sad, and D. Sullivan. On the dynamics of rational maps. Ann. Sci. Ec. Norm. Super. (4), 16:193-217, 1983. https://doi.org/10.24033/asens.1446 pathdoi:10.24033/asens.1446.
37. X. Menguy. Noncollapsing examples with positive Ricci curvature and infinite topological type. Geom. Funct. Anal., 10(3):600-627, 2000. https://doi.org/10.1007/PL00001632 pathdoi:10.1007/PL00001632.
38. A. Mori. On an absolute constant in the theory of quasi-conformal mappings. J. Math. Soc. Japan, 8:156-166, 1956. https://doi.org/10.2969/jmsj/00820156 pathdoi:10.2969/jmsj/00820156.
39. K. Nagano. Asymptotic topological regularity of (mathrmCAT(0)) spaces. Ann. Global Anal. Geom., 61(2):427-457, 2022. https://doi.org/10.1007/s10455-021-09820-2 pathdoi:10.1007/s10455-021-09820-2.
40. R. Osserman. From Schwarz to Pick to Ahlfors and beyond. Notices Amer. Math. Soc., 46(8):868-873, 1999.
41. P. Pansu. Dimension conforme et sph`ere `a l'infini des varietes `a courbure negative. (Conformal dimension and the ideal boundary of manifolds with negative curvature). Ann. Acad. Sci. Fenn., Ser. A I, Math., 14(2):177-212, 1989. https://doi.org/10.5186/aasfm.1989.1424 pathdoi:10.5186/aasfm.1989.1424.
42. A. Papadopoulos. Nicolas-Auguste Tissot: a link between cartography and quasiconformal theory. Arch. Hist. Exact Sci., 71(4):319-336, 2017. https://doi.org/10.1007/s00407-016-0186-z pathdoi:10.1007/s00407-016-0186-z.
43. A. Papadopoulos. Quasiconformal mappings, from Ptolemy's geography to the work of Teichmuller. In Uniformization, Riemann-Hilbert correspondence, Calabi-Yau manifolds & Picard-Fuchs equations, volume 42 of Adv. Lect. Math. (ALM), pages 237-314. Int. Press, Somerville, MA, 2018.
44. A. Papadopoulos. A note on Nicolas-Auguste Tissot: at the origin of quasiconformal mappings. In Handbook of Teichmuller theory. Vol. VII, volume 30 of IRMA Lect. Math. Theor. Phys., pages 289-299. Eur. Math. Soc., Zurich, 2020.
45. G. Perelman. Manifolds of positive Ricci curvature with almost maximal volume. J. Amer. Math. Soc., 7(2):299-305, 1994. https://doi.org/10.2307/2152760 pathdoi:10.2307/2152760.
46. A. Petrunin. Alexandrov meets Lott-Villani-Sturm. Munster J. Math., 4:53-64, 2011.
47. S. Rickman. On the number of omitted values of entire quasiregular mappings. J. Analyse Math., 37:100-117, 1980. https://doi.org/10.1007/BF02797681 pathdoi:10.1007/BF02797681.
48. K.-T. Sturm. On the geometry of metric measure spaces. I. Acta Math., 196(1):65-131, 2006. https://doi.org/10.1007/s11511-006-0002-8 pathdoi:10.1007/s11511-006-0002-8.
49. D. Sullivan. Quasiconformal homeomorphisms in dynamics, topology, and geometry. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pages 1216-1228. Amer. Math. Soc., Providence, RI, 1987.
50. A. Tissot. Memoire sur la representation des surfaces et les projections des cartes geographiques. Nouv. Ann. (2), 18:385-397, 1879.
51. C. Villani. Optimal transport, volume 338 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009. Old and new. https://doi.org/10.1007/978-3-540-71050-9 pathdoi:10.1007/978-3-540-71050-9.
52. S.-H. Zhu. A finiteness theorem for Ricci curvature in dimension three. J. Differential Geom., 37(3):711-727, 1993. URL: http://projecteuclid.org/euclid.jdg/1214453906.
53. V. A. Zorich. Homeomorphism of quasiconformal space mappings. Sov. Math. Dokl., 8:1039-1042, 1967
2. L. V. Ahlfors. Lectures on quasiconformal mappings, volume 38 of University Lecture Series. American Mathematical Society, Providence, RI, second edition, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. https://doi.org/10.1090/ulect/038 pathdoi:10.1090/ulect/038.
3. V. Alberge and A. Papadopoulos. A commentary on Lavrentieff's paper \"Sur une classe de representations continues\". In Handbook of Teichmuller theory. Vol. VII, volume 30 of IRMA Lect. Math. Theor. Phys., pages 441-451. Eur. Math. Soc., Zurich, 2020.
4. V. Alberge and A. Papadopoulos. On five papers by Herbert Grotzsch. In Handbook of Teichmuller theory. Vol. VII, volume 30 of IRMA Lect. Math. Theor. Phys., pages 393-415. Eur. Math. Soc., Zurich, 2020.
5. A. D. Aleksandrov. Vnutrennyaya Geometriya Vypuklyh Poverhnosteui. OGIZ, Moscow-Leningrad, 1948.
6. L. Ambrosio. Calculus, heat flow and curvature-dimension bounds in metric measure spaces. In Proceedings of the International Congress of Mathematicians-Rio de Janeiro 2018. Vol. I. Plenary lectures, pages 301-340. World Sci. Publ., Hackensack, NJ, 2018.
7. L. Ambrosio, S. Di Marino, and N. Gigli. Perimeter as relaxed Minkowski content in metric measure spaces. Nonlinear Anal., 153:78-88, 2017. https://doi.org/10.1016/j.na.2016.03.010 pathdoi:10.1016/j.na.2016.03.010.
8. L. Ambrosio, A. Mondino, and G. Savare. Nonlinear diffusion equations and curvature conditions in metric measure spaces. Mem. Amer. Math. Soc., 262(1270):v+121, 2019. https://doi.org/10.1090/memo/1270 pathdoi:10.1090/memo/1270.
9. Z. M. Balogh and A. Kristaly. Sharp isoperimetric and sobolev inequalities in spaces with nonnegative ricci curvature. Mathematische Annalen, March 2022. https://doi.org/10.1007/s00208-022-02380-1 pathdoi:10.1007/s00208-022-02380-1.
10. D. Burago, Y. Burago, and S. Ivanov. A course in metric geometry, volume 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. https://doi.org/10.1090/gsm/033 pathdoi:10.1090/gsm/033.
11. G. Carron. Euclidean volume growth for complete Riemannian manifolds. Milan J. Math., 88(2):455-478, 2020. https://doi.org/10.1007/s00032-020-00321-8 pathdoi:10.1007/s00032-020-00321-8.
12. F. Cavalletti and E. Milman. The globalization theorem for the curvature-dimension condition. Invent. Math., 226(1):1-137, 2021. https://doi.org/10.1007/s00222-021-01040-6 pathdoi:10.1007/s00222-021-01040-6.
13. J. Cheeger and T. H. Colding. On the structure of spaces with Ricci curvature bounded below. I. J. Differential Geom., 46(3):406-480, 1997. URL: http://projecteuclid.org/euclid.jdg/1214459974.
14. J. Deng. Curvature-dimension condition meets Gromov's n-volumic scalar curvature. SIGMA Symmetry Integrability Geom. Methods Appl., 17:Paper No. 013, 20, 2021. https://doi.org/10.3842/SIGMA.2021.013 pathdoi:10.3842/SIGMA.2021.013.
15. J. Deng. Enlargeable length-structure and scalar curvatures. Ann. Global Anal. Geom., 60(2):217-230, 2021. https://doi.org/10.1007/s10455-021-09772-7 pathdoi:10.1007/s10455-021-09772-7.
16. J. Deng. Foliated Positive Scalar Curvature. PhD thesis, University of Goettingen, 2021. https://doi.org/10.53846/goediss-8930 pathdoi:10.53846/goediss-8930.
17. Q. Deng. Holder continuity of tangent cones in RCD(K,N) spaces and applications to non-branching. arXiv e-prints, page arXiv:2009.07956, September 2020. http://arxiv.org/abs/2009.07956 patharXiv:2009.07956.
18. M. Erbar, K. Kuwada, and K.-T. Sturm. On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces. Invent. Math., 201(3):993-1071, 2015. https://doi.org/10.1007/s00222-014-0563-7 pathdoi:10.1007/s00222-014-0563-7.
19. R. Fehlmann and M. Vuorinen. Mori's theorem for n-dimensional quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math., 13(1):111-124, 1988. https://doi.org/10.5186/aasfm.1988.1304 pathdoi:10.5186/aasfm.1988.1304.
20. F. P. Gardiner and N. Lakic. Quasiconformal Teichmuller theory, volume 76 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000. https://doi.org/10.1090/surv/076 pathdoi:10.1090/surv/076.
21. F. W. Gehring, G. J. Martin, and B. P. Palka. An introduction to the theory of higher-dimensional quasiconformal mappings, volume 216 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2017. https://doi.org/10.1090/surv/216 pathdoi:10.1090/surv/216.
22. F. W. Gehring and J. Vaisala. Hausdorff dimension and quasiconformal mappings. J. Lond. Math. Soc. (2), 6:504-512, 1973. https://doi.org/10.1112/jlms/s2-6.3.504 pathdoi:10.1112/jlms/s2-6.3.504.
23. N. Gigli. Nonsmooth differential geometry - an approach tailored for spaces with Ricci curvature bounded from below. Mem. Amer. Math. Soc., 251(1196):v+161, 2018. https://doi.org/10.1090/memo/1196 pathdoi:10.1090/memo/1196.
24. N. Gigli and E. Pasqualetto. Lectures on nonsmooth differential geometry, volume 2 of SISSA Springer Series. Springer, Cham, 2020. https://doi.org/10.1007/978-3-030-38613-9 pathdoi:10.1007/978-3-030-38613-9.
25. H. Grotzsch. Uber die Verzerrung bei schlichten nichtkonformen Abbildungen und uber eine damit zusammenhangende Erweiterung des Picardschen Satzes., 1928.
26. H. Grotzsch. Uber moglichst konforme Abbildungen von schlichten Bereichen., 1932.
27. D. X. Gu, W. Zeng, L. M. Lui, F. Luo, and S.-T. Yau. Recent development of computational conformal geometry. In Fifth International Congress of Chinese Mathematicians. Part 1, 2, volume 2 of AMS/IP Stud. Adv. Math., 51, pt. 1, pages 515-560. Amer. Math. Soc., Providence, RI, 2012.
28. J. Heinonen and P. Koskela. Quasiconformal maps in metric spaces with controlled geometry. Acta Math., 181(1):1-61, 1998. https://doi.org/10.1007/BF02392747 pathdoi:10.1007/BF02392747.
29. S. Honda. New differential operator and noncollapsed RCD spaces. Geom. Topol., 24(4):2127-2148, 2020. https://doi.org/10.2140/gt.2020.24.2127 pathdoi:10.2140/gt.2020.24.2127.
30. H. Huang and X.-T. Huang. Almost splitting maps, transformation theorems and smooth fibration theorems. arXiv e-prints, page arXiv:2207.10029, July 2022. http://arxiv.org/abs/2207.10029 patharXiv:2207.10029.
31. V. Kapovitch and A. Mondino. On the topology and the boundary of N-dimensional mathrmRCD(K,N) spaces. Geom. Topol., 25(1):445-495, 2021. https://doi.org/10.2140/gt.2021.25.445 pathdoi:10.2140/gt.2021.25.445.
32. S. Keith and X. Zhong. The Poincare inequality is an open ended condition. Ann. of Math. (2), 167(2):575-599, 2008. https://doi.org/10.4007/annals.2008.167.575 pathdoi:10.4007/annals.2008.167.575.
33. M. Lavrentieff. On a class of continuous representations. In Handbook of Teichmuller theory. Volume VII, pages 417-439. Berlin: European Mathematical Society (EMS), 2020. https://doi.org/10.4171/203-1/20 pathdoi:10.4171/203-1/20.
34. J. Lott and C. Villani. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2), 169(3):903-991, 2009. https://doi.org/10.4007/annals.2009.169.903 pathdoi:10.4007/annals.2009.169.903.
35. L. M. Lui, T. W. Wong, W. Zeng, X. Gu, P. M. Thompson, T. F. Chan, and S.-T. Yau. A survey on recent development in computational quasi-conformal geometry and its applications. In Fifth International Congress of Chinese Mathematicians. Part 1, 2, volume 2 of AMS/IP Stud. Adv. Math., 51, pt. 1, pages 697-717. Amer. Math. Soc., Providence, RI, 2012. https://doi.org/10.3934/ipi.2010.4.311 pathdoi:10.3934/ipi.2010.4.311.
36. R. Mane, P. Sad, and D. Sullivan. On the dynamics of rational maps. Ann. Sci. Ec. Norm. Super. (4), 16:193-217, 1983. https://doi.org/10.24033/asens.1446 pathdoi:10.24033/asens.1446.
37. X. Menguy. Noncollapsing examples with positive Ricci curvature and infinite topological type. Geom. Funct. Anal., 10(3):600-627, 2000. https://doi.org/10.1007/PL00001632 pathdoi:10.1007/PL00001632.
38. A. Mori. On an absolute constant in the theory of quasi-conformal mappings. J. Math. Soc. Japan, 8:156-166, 1956. https://doi.org/10.2969/jmsj/00820156 pathdoi:10.2969/jmsj/00820156.
39. K. Nagano. Asymptotic topological regularity of (mathrmCAT(0)) spaces. Ann. Global Anal. Geom., 61(2):427-457, 2022. https://doi.org/10.1007/s10455-021-09820-2 pathdoi:10.1007/s10455-021-09820-2.
40. R. Osserman. From Schwarz to Pick to Ahlfors and beyond. Notices Amer. Math. Soc., 46(8):868-873, 1999.
41. P. Pansu. Dimension conforme et sph`ere `a l'infini des varietes `a courbure negative. (Conformal dimension and the ideal boundary of manifolds with negative curvature). Ann. Acad. Sci. Fenn., Ser. A I, Math., 14(2):177-212, 1989. https://doi.org/10.5186/aasfm.1989.1424 pathdoi:10.5186/aasfm.1989.1424.
42. A. Papadopoulos. Nicolas-Auguste Tissot: a link between cartography and quasiconformal theory. Arch. Hist. Exact Sci., 71(4):319-336, 2017. https://doi.org/10.1007/s00407-016-0186-z pathdoi:10.1007/s00407-016-0186-z.
43. A. Papadopoulos. Quasiconformal mappings, from Ptolemy's geography to the work of Teichmuller. In Uniformization, Riemann-Hilbert correspondence, Calabi-Yau manifolds & Picard-Fuchs equations, volume 42 of Adv. Lect. Math. (ALM), pages 237-314. Int. Press, Somerville, MA, 2018.
44. A. Papadopoulos. A note on Nicolas-Auguste Tissot: at the origin of quasiconformal mappings. In Handbook of Teichmuller theory. Vol. VII, volume 30 of IRMA Lect. Math. Theor. Phys., pages 289-299. Eur. Math. Soc., Zurich, 2020.
45. G. Perelman. Manifolds of positive Ricci curvature with almost maximal volume. J. Amer. Math. Soc., 7(2):299-305, 1994. https://doi.org/10.2307/2152760 pathdoi:10.2307/2152760.
46. A. Petrunin. Alexandrov meets Lott-Villani-Sturm. Munster J. Math., 4:53-64, 2011.
47. S. Rickman. On the number of omitted values of entire quasiregular mappings. J. Analyse Math., 37:100-117, 1980. https://doi.org/10.1007/BF02797681 pathdoi:10.1007/BF02797681.
48. K.-T. Sturm. On the geometry of metric measure spaces. I. Acta Math., 196(1):65-131, 2006. https://doi.org/10.1007/s11511-006-0002-8 pathdoi:10.1007/s11511-006-0002-8.
49. D. Sullivan. Quasiconformal homeomorphisms in dynamics, topology, and geometry. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pages 1216-1228. Amer. Math. Soc., Providence, RI, 1987.
50. A. Tissot. Memoire sur la representation des surfaces et les projections des cartes geographiques. Nouv. Ann. (2), 18:385-397, 1879.
51. C. Villani. Optimal transport, volume 338 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009. Old and new. https://doi.org/10.1007/978-3-540-71050-9 pathdoi:10.1007/978-3-540-71050-9.
52. S.-H. Zhu. A finiteness theorem for Ricci curvature in dimension three. J. Differential Geom., 37(3):711-727, 1993. URL: http://projecteuclid.org/euclid.jdg/1214453906.
53. V. A. Zorich. Homeomorphism of quasiconformal space mappings. Sov. Math. Dokl., 8:1039-1042, 1967