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Reshetnyak's Theory of Subharmonic Metrics pp 47–145Cite as

An Introduction to Reshetnyak’s Theory of Subharmonic Distances

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Abstract

The aim of the present text is to provide some basics around Reshetnyak’s theory of subharmonic distances on surfaces, together with an overview of the main results. While subharmonic distances are often confused with two-dimensional manifolds of bounded curvature, we present them as a complete autonomous theory. In turn, there is no specific prerequisite for the present text.

Keywords

  • Subharmonic metrics
  • Alexandrov surfaces
  • Conformal metrics
  • Curves of bounded rotation
  • Plane potential theory
  • Riemann surfaces

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Notes

  1. 1.

    Relaxing the curvature bound assumption would be too general, see, e.g., [33]. See [30] if the condition for the limit metric to have the topology of a surface is relaxed.

  2. 2.

    The term “Alexandrov surface” is equivocal in the literature. Sometimes, it means a two-dimensional manifold of bounded curvature; sometimes it means a metric surface with a curvature bound in the sense of Alexandrov and sometimes a metric surface with a lower curvature bound. In turn, we will not use further this terminology.

  3. 3.

    There is a slight difference in notation with the corresponding formula in [84] (Chap. 7) because Reshetnyak is considering λ = e−2p(ω)+h.

  4. 4.

    This is the actual meaning of regular measure and is different from the regular measures studied in Chap. 13, see Remark 4.17.

  5. 5.

    Intrinsic metric space may also be called inner or internal.

  6. 6.

    This theorem is called Metrics Convergence Theorem in Reshetnyak articles, as the definitions do not fit with ours, see the introduction.

  7. 7.

    We are using the notation ψ here to fit with the one from [90] (Chap. 13), but it is not the function used in Lemma 4.82 or in the proof of the Localization Theorem 4.85.

  8. 8.

    The following alternative terminologies are also used for contracting mapping: inextensible, non-extensible, non-expansive, short, and metric.

  9. 9.

    On page 96 in [92], “Theorem 6.3.2” should be read “Theorem 6.2.2.”

  10. 10.

    The following argument was suggested to the author by Alexander Lytchak.

  11. 11.

    Here and in similar statements, the distance that is approximated is the induced intrinsic distance over the neighborhood.

  12. 12.

    Here also, the distance over the neighborhood is the induced intrinsic distance. Actually, a stronger result is stated in [84] (Chap. 7), namely that the interior of any neighborhood homeomorphic to a closed disc has isothermal coordinates, see Corollary 4.160. In [84] (Chap. 7), a stronger (smooth) version of Theorem 4.165 is stated and used.

  13. 13.

    In [11], what we are calling turn was translated by rotation; see the Introduction of the present article.

  14. 14.

    The fact that the angles coincide is implicit in §8 of [90] (Chap. 13).

  15. 15.

    Actually, (4.28) is stated only for regular arcs. It is easy to extend to arcs of the class Δ, i.e., arcs having semitangent at each point, see [92] before Theorem 8.1.7. Anyway, when the two Borelian measures coincide on the Euclidean balls, they are equal, see, e.g., [56], [70, 2.20].

  16. 16.

    In Theorems 4.171 and 4.173, weak convergence is for functions with a same compact support, see Remark 4.60.

  17. 17.

    Functions such that their logarithm is subharmonic are called PL, see, e.g., [23, 79].

References

  1. Adamowicz, T., Veronelli, G.: Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces. Calc. Var. Partial Differential Equations 61(1), Paper No. 2, 30 (2022). https://doi.org/10.1007/s00526-021-02109-z

  2. Ahlfors, L.V.: Lectures on quasiconformal mappings, University Lecture Series, vol. 38, second edn. American Mathematical Society, Providence, RI (2006). https://doi.org/10.1090/ulect/038. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard

  3. Ahlfors, L.V., Sario, L.: Riemann surfaces. Princeton Mathematical Series, No. 26. Princeton University Press, Princeton, N.J. (1960)

    Google Scholar 

  4. Alexander, S., Kapovitch, V., Petrunin, A.: Alexandrov geometry: preliminary version no. 1. arXiv e-prints arXiv:1903.08539 (2019)

    Google Scholar 

  5. Alexander, S., Kapovitch, V., Petrunin, A.: An invitation to Alexandrov geometry. SpringerBriefs in Mathematics. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-05312-3. CAT(0) spaces

  6. Alexandrov, A.D.: Über eine Verallgemeinerung der Riemannschen Geometrie. Schr. Forschungsinst. Math. 1, 33–84 (1957)

    Google Scholar 

  7. Alexandrov, A.D.: A. D. Alexandrov selected works. Part II. Chapman & Hall/CRC, Boca Raton, FL (2006). Intrinsic geometry of convex surfaces, Edited by S. S. Kutateladze, Translated from the Russian by S. Vakhrameyev

    Google Scholar 

  8. Alexandrov, A.D.: On the surfaces representable as difference of convex functions [translation of mr0048059]. Sib. Èlektron. Mat. Izv. 9, 360–376 (2012)

    Google Scholar 

  9. Alexandrov, A.D., Reshetnyak, Y.G.: General theory of irregular curves, Mathematics and its Applications (Soviet Series), vol. 29. Kluwer Academic Publishers Group, Dordrecht (1989). https://doi.org/10.1007/978-94-009-2591-5. Translated from the Russian by L. Ya. Yuzina

  10. Alexandrov, A.D., Zalgaller, V.A. (eds.): Two-dimensional manifolds of bounded curvature. Proceedings of the Steklov Institute of Mathematics, No. 76 (1965). American Mathematical Society, Providence, R.I. (1965). Translated from the Russian by J. M. Danskin

    Google Scholar 

  11. Alexandrov, A.D., Zalgaller, V.A.: Intrinsic geometry of surfaces. Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, Vol. 15. American Mathematical Society, Providence, R.I. (1967)

    Google Scholar 

  12. Ambrosio, L., Bertrand, J.: On the regularity of Alexandrov surfaces with curvature bounded below. Anal. Geom. Metr. Spaces 4(1), 282–287 (2016). https://doi.org/10.1515/agms-2016-0012

  13. Ambrosio, L., Bertrand, J.: DC calculus. Math. Z. 288(3–4), 1037–1080 (2018). https://doi.org/10.1007/s00209-017-1926-8

  14. Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)

    Google Scholar 

  15. Ambrosio, L., Tilli, P.: Topics on analysis in metric spaces, Oxford Lecture Series in Mathematics and its Applications, vol. 25. Oxford University Press, Oxford (2004)

    Google Scholar 

  16. Armitage, D.H., Gardiner, S.J.: Classical potential theory. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London (2001). https://doi.org/10.1007/978-1-4471-0233-5

  17. Arsove, M.G.: Functions representable as differences of subharmonic functions. Trans. Amer. Math. Soc. 75, 327–365 (1953). https://doi.org/10.2307/1990736

  18. Ash, R.B.: Measure, integration, and functional analysis. Academic Press, New York-London (1972)

    Google Scholar 

  19. Bak, J., Newman, D.J.: Complex analysis, third edn. Undergraduate Texts in Mathematics. Springer, New York (2010). https://doi.org/10.1007/978-1-4419-7288-0

  20. Bartle, R.G., Joichi, J.T.: The preservation of convergence of measurable functions under composition. Proc. Amer. Math. Soc. 12, 122–126 (1961). https://doi.org/10.2307/2034137

  21. Bartolucci, D., Castorina, D.: On a singular Liouville-type equation and the Alexandrov isoperimetric inequality. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 35–64 (2019)

    Google Scholar 

  22. Beardon, A.F.: A primer on Riemann surfaces, London Mathematical Society Lecture Note Series, vol. 78. Cambridge University Press, Cambridge (1984)

    Google Scholar 

  23. Beckenbach, E.F., Radó, T.: Subharmonic functions and surfaces of negative curvature. Trans. Amer. Math. Soc. 35(3), 662–674 (1933). https://doi.org/10.2307/1989854

  24. Bonahon, F.: Low-dimensional geometry, Student Mathematical Library, vol. 49. American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ (2009). https://doi.org/10.1090/stml/049. From Euclidean surfaces to hyperbolic knots, IAS/Park City Mathematical Subseries

  25. Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin (1999). https://doi.org/10.1007/978-3-662-12494-9

  26. Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry, Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence, RI (2001). https://doi.org/10.1090/gsm/033

  27. Burago, Y.: On proportional approximation of a metric. Trudy Mat. Inst. Steklov. 76, 120–123 (1965). English translation in [10]

    Google Scholar 

  28. Burago, Y.: Bi-Lipschitz-equivalent Aleksandrov surfaces. II. Algebra i Analiz 16(6), 28–52 (2004). https://doi.org/10.1090/S1061-0022-05-00885-X

  29. Burago, Y., Buyalo, S.: Metrics with upper-bounded curvature on 2-polyhedra. II. Algebra i Analiz 10(4), 62–112 (1998)

    Google Scholar 

  30. Burago, Y.D.: The closure of a class of manifolds with bounded curvature. Trudy Mat. Inst. Steklov. 76, 141–147 (1965). English translation in [10]

  31. do Carmo, M.P.: Differential geometry of curves & surfaces. Dover Publications, Inc., Mineola, NY (2016). Revised & updated second edition of [ MR0394451]

    Google Scholar 

  32. Cartan, H.: Théorie du potentiel Newtonien: énergie, capacité, suites de potentials. Bull. Soc. Math. France 73, 74–106 (1945)

    Google Scholar 

  33. Cassorla, M.: Approximating compact inner metric spaces by surfaces. Indiana Univ. Math. J. 41(2), 505–513 (1992). https://doi.org/10.1512/iumj.1992.41.41029

  34. Chen, J., Li, Y.: Uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature (2022). https://doi.org/10.48550/arXiv.2208.05620

  35. Chern, S.s.: An elementary proof of the existence of isothermal parameters on a surface. Proc. Amer. Math. Soc. 6, 771–782 (1955). https://doi.org/10.2307/2032933

  36. Chern, S.s., Hartman, P., Wintner, A.: On isothermic coordinates. Comment. Math. Helv. 28, 301–309 (1954). https://doi.org/10.1007/BF02566936

  37. Chirka, E.M.: Potentials on a compact Riemann surface. Tr. Mat. Inst. Steklova 301(Kompleksnyı̆ Analiz, Matematicheskaya Fizika i Prilozheniya), 287–319 (2018). https://doi.org/10.1134/S0371968518020218

  38. Chowdhury, S., Hu, H., Romney, M., Tsou, A.: On cat(k) surfaces (2022)

    Google Scholar 

  39. Creutz, P., Romney, M.: Triangulating metric surfaces. Proc. Lond. Math. Soc. (2022). https://doi.org/10.1112/plms.12486

  40. Dacorogna, B.: Introduction to the calculus of variations, second edn. Imperial College Press, London (2009). Translated from the 1992 French original

    Google Scholar 

  41. Debin, C.: A compactness theorem for surfaces with bounded integral curvature. J. Inst. Math. Jussieu 19(2), 597–645 (2020). https://doi.org/10.1017/s1474748018000154

  42. Demailly, J.P.: Complex analytic and differential geometry. Author’s website (2012)

    Google Scholar 

  43. Donaldson, S.: Riemann surfaces, Oxford Graduate Texts in Mathematics, vol. 22. Oxford University Press, Oxford (2011). https://doi.org/10.1093/acprof:oso/9780198526391.001.0001

  44. Dudley, R.M.: On second derivatives of convex functions. Math. Scand. 41(1), 159–174 (1977). https://doi.org/10.7146/math.scand.a-11710

  45. Dudley, R.M.: Acknowledgment of priority: “On second derivatives of convex functions” [Math. Scand. 41 (1977), no. 1, 159–174; MR 58 #2250]. Math. Scand. 46(1), 61 (1980). https://doi.org/10.7146/math.scand.a-11851

  46. Falconer, K.: Fractal geometry, third edn. John Wiley & Sons, Ltd., Chichester (2014). Mathematical foundations and applications

    Google Scholar 

  47. Falconer, K.J.: The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  48. Farb, B., Margalit, D.: A primer on mapping class groups, Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton, NJ (2012)

    Google Scholar 

  49. Farkas, H.M., Kra, I.: Riemann surfaces, Graduate Texts in Mathematics, vol. 71, second edn. Springer-Verlag, New York (1992). https://doi.org/10.1007/978-1-4612-2034-3

  50. Folland, G.B.: Real analysis, second edn. Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York (1999). Modern techniques and their applications, A Wiley-Interscience Publication

    Google Scholar 

  51. Forster, O.: Lectures on Riemann surfaces, Graduate Texts in Mathematics, vol. 81. Springer-Verlag, New York (1991). Translated from the 1977 German original by Bruce Gilligan, Reprint of the 1981 English translation

    Google Scholar 

  52. Gardiner, F.P., Lakic, N.: Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs, vol. 76. American Mathematical Society, Providence, RI (2000)

    Google Scholar 

  53. Hartman, P.: On functions representable as a difference of convex functions. Pacific J. Math. 9, 707–713 (1959). http://projecteuclid.org/euclid.pjm/1103039111

  54. Hartman, P., Wintner, A.: On uniform Dini conditions in the theory of linear partial differential equations of elliptic type. Amer. J. Math. 77, 329–354 (1955). https://doi.org/10.2307/2372534

  55. Hayman, W.K., Kennedy, P.B.: Subharmonic functions. Vol. I. Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York (1976). London Mathematical Society Monographs, No. 9

    Google Scholar 

  56. Hoffmann-Jørgensen, J.: Measures which agree on balls. Math. Scand. 37(2), 319–326 (1975). https://doi.org/10.7146/math.scand.a-11610

  57. Hörmander, L.: The analysis of linear partial differential operators. II. Classics in Mathematics. Springer-Verlag, Berlin (2005). https://doi.org/10.1007/b138375. Differential operators with constant coefficients, Reprint of the 1983 original

  58. Huber, A.: Zum potentialtheoretischen Aspekt der Alexandrowschen Flächentheorie. Comment. Math. Helv. 34, 99–126 (1960). https://doi.org/10.1007/BF02565931

  59. Hulin, D., Troyanov, M.: Prescribing curvature on open surfaces. Math. Ann. 293(2), 277–315 (1992). https://doi.org/10.1007/BF01444716

  60. Imomkulov, S.A.: Twice differentiability of subharmonic functions. Izv. Ross. Akad. Nauk Ser. Mat. 56(4), 877–888 (1992). https://doi.org/10.1070/IM1993v041n01ABEH002184

  61. Izmestiev, I.: A simple proof of an isoperimetric inequality for Euclidean and hyperbolic cone-surfaces. Differential Geom. Appl. 43, 95–101 (2015). https://doi.org/10.1016/j.difgeo.2015.09.007

  62. Jost, J.: Postmodern analysis, third edn. Universitext. Springer-Verlag, Berlin (2005)

    Google Scholar 

  63. Jost, J.: Compact Riemann surfaces, third edn. Universitext. Springer-Verlag, Berlin (2006). https://doi.org/10.1007/978-3-540-33067-7. An introduction to contemporary mathematics

  64. Jost, J.: Riemannian geometry and geometric analysis, sixth edn. Universitext. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21298-7

  65. Kapovich, M.: Hyperbolic manifolds and discrete groups. Modern Birkhäuser Classics. Birkhäuser Boston, Ltd., Boston, MA (2009). https://doi.org/10.1007/978-0-8176-4913-5. Reprint of the 2001 edition

  66. Kokarev, G.: Curvature and bubble convergence of harmonic maps. J. Geom. Anal. 23(3), 1058–1077 (2013). https://doi.org/10.1007/s12220-011-9273-1

  67. Landkof, N.S.: Foundations of modern potential theory. Springer-Verlag, New York-Heidelberg (1972). Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180

    Google Scholar 

  68. Lehto, O., Virtanen, K.I.: Quasiconformal mappings in the plane, second edn. Springer-Verlag, New York-Heidelberg (1973). Translated from the German by K. W. Lucas, Die Grundlehren der mathematischen Wissenschaften, Band 126

    Google Scholar 

  69. Lytchak, A., Wenger, S.: Isoperimetric characterization of upper curvature bounds. Acta Math. 221(1), 159–202 (2018). https://doi.org/10.4310/ACTA.2018.v221.n1.a5

  70. Mattila, P.: Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995). https://doi.org/10.1017/CBO9780511623813. Fractals and rectifiability

  71. Men’shov, D.E.: Sur une généralisation d’un théorème de M. H. Bohr. Rec. Math. Moscou, n. Ser. 2, 339–354 (1937)

    Google Scholar 

  72. Milnor, J.W.: On the total curvature of knots. Ann. of Math. (2) 52, 248–257 (1950). https://doi.org/10.2307/1969467

  73. Natanson, I.P.: Theory of functions of a real variable. Frederick Ungar Publishing Co., New York (1955). Translated by Leo F. Boron with the collaboration of Edwin Hewitt

    Google Scholar 

  74. Nikolaev, I.: Foliations on surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 41. Springer-Verlag, Berlin (2001). https://doi.org/10.1007/978-3-662-04524-4. With a foreword by Idel Bronshteyn and Chapter 16 by B. Piccoli

  75. Papadopoulos, A.: Metric spaces, convexity and non-positive curvature, IRMA Lectures in Mathematics and Theoretical Physics, vol. 6, second edn. European Mathematical Society (EMS), Zürich (2014). https://doi.org/10.4171/132

  76. Perelman, G.: DC structure on Alexandrov space. Preprint, 1994

    Google Scholar 

  77. Pogorelov, A.V.: Extrinsic geometry of convex surfaces. Translations of Mathematical Monographs, Vol. 35. American Mathematical Society, Providence, R.I. (1973). Translated from the Russian by Israel Program for Scientific Translations

    Google Scholar 

  78. Pokorný, D., Rataj, J., Zajíček, L.: On the structure of WDC sets. Math. Nachr. 292(7), 1595–1626 (2019). https://doi.org/10.1002/mana.201700253

  79. Radó, T.: On the problem of Plateau. Subharmonic functions. Springer-Verlag, New York-Heidelberg (1971). Reprint

    Google Scholar 

  80. Radon, J.: Über die Randwertaufgaben beim logarithmischen Potential. Anzeiger Wien 56, 190 (1919)

    Google Scholar 

  81. Ransford, T.: Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28. Cambridge University Press, Cambridge (1995). https://doi.org/10.1017/CBO9780511623776

  82. Rao, M.: Brownian motion and classical potential theory. Matematisk Institut, Aarhus University, Aarhus (1977). Lecture Notes Series, No. 47

    Google Scholar 

  83. Reshetnyak, Y.G.: Study of manifolds of bounded curvature using isothermal coordinates. Izvestiia Sibirskogo otdeleniia Akademii nauk SSSR (1959)

    Google Scholar 

  84. Reshetnyak, Y.G.: Isothermal coordinates on manifolds of bounded curvature I. Sibirsk. Mat. Ž. 1, 88–116 (1960)

    Google Scholar 

  85. Reshetnyak, Y.G.: Isothermal coordinates on manifolds of bounded curvature II. Sibirsk. Mat. Ž. 1, 248–276 (1960)

    Google Scholar 

  86. Reshetnyak, Y.G.: On a special mapping of a cone onto a polyhedron. Mat. Sb. (N.S.) 53 (95), 39–52 (1961)

    Google Scholar 

  87. Reshetnyak, Y.G.: On isoperimetric property of two dimensional manifolds with curvature bounded from above by K. Vestnik Leningrad. Univ. 16(19), 58–76 (1961)

    Google Scholar 

  88. Reshetnyak, Y.G.: On a special mapping of a cone in a manifold of bounded curvature. Sibirsk. Mat. Ž. 3, 256–272 (1962)

    Google Scholar 

  89. Reshetnyak, Y.G.: Arc length in a manifold of bounded curvature with an isothermal metric. Sibirsk. Mat. Ž. 4, 212–226 (1963)

    Google Scholar 

  90. Reshetnyak, Y.G.: Turn of curves in a manifold of bounded curvature with isothermal metric. Sibirsk. Mat. Ž. 4, 870–911 (1963)

    Google Scholar 

  91. Reshetnyak, Y.G.: Non-expansive maps in a space of curvature no greater than k. Sibirsk. Mat. Ž. 9, 918–927 (1968)

    Google Scholar 

  92. Reshetnyak, Y.G.: Two-dimensional manifolds of bounded curvature. In: Geometry, IV, Encyclopaedia Math. Sci., vol. 70, pp. 3–163, 245–250. Springer, Berlin (1993)

    Google Scholar 

  93. Reshetnyak, Y.G.: On the conformal representation of Alexandrov surfaces. In: Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat., vol. 83, pp. 287–304. Univ. Jyväskylä, Jyväskylä (2001)

    Google Scholar 

  94. Reshetnyak, Y.G.: The theory of curves in differential geometry from the point of view of the theory of functions of a real variable. Uspekhi Mat. Nauk 60(6(366)), 157–174 (2005). https://doi.org/10.1070/RM2005v060n06ABEH004286

  95. Reshetnyak, Y.G.: Personal communication (2021)

    Google Scholar 

  96. Richard, T.: Canonical smoothing of compact Aleksandrov surfaces via Ricci flow. Ann. Sci. Éc. Norm. Supér. (4) 51(2), 263–279 (2018). https://doi.org/10.24033/asens.2356

  97. Roberts, A.W., Varberg, D.E.: Convex functions. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1973). Pure and Applied Mathematics, Vol. 57

    Google Scholar 

  98. Royden, H.L.: Function theory on compact Riemann surfaces. J. Analyse Math. 18, 295–327 (1967). https://doi.org/10.1007/BF02798051

  99. Rudin, W.: Real and complex analysis, third edn. McGraw-Hill Book Co., New York (1987)

    Google Scholar 

  100. Schwartz, L.: Théorie des distributions. Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée. Hermann, Paris (1966)

    Google Scholar 

  101. Spivak, M.: A comprehensive introduction to differential geometry. Vol. III, second edn. Publish or Perish, Inc., Wilmington, Del. (1979)

    Google Scholar 

  102. Stratilatova, M.B.: Area in two-dimensional manifolds of bounded curvature as Hausdorff measure. Vestnik Leningrad. Univ. 17(13), 56–60 (1962)

    Google Scholar 

  103. Sullivan, J.M.: Curves of finite total curvature. In: Discrete differential geometry, Oberwolfach Semin., vol. 38, pp. 137–161. Birkhäuser, Basel (2008)

    Google Scholar 

  104. Szpilrajn, E.: Remarques sur les fonctions sousharmoniques. Ann. of Math. (2) 34(3), 588–594 (1933). https://doi.org/10.2307/1968179

  105. Troyanov, M.: Les surfaces euclidiennes à singularités coniques. Enseign. Math. (2) 32(1–2), 79–94 (1986)

    Google Scholar 

  106. Troyanov, M.: Un principe de concentration-compacité pour les suites de surfaces riemanniennes. Ann. Inst. H. Poincaré Anal. Non Linéaire 8(5), 419–441 (1991). https://doi.org/10.1016/S0294-1449(16)30255-4

  107. Troyanov, M.: Surfaces riemanniennes à singularités simples. In: Differential geometry. Part 2: Geometry in mathematical physics and related topics. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8–28, 1990, pp. 619–628. Providence, RI: American Mathematical Society (1993)

    Google Scholar 

  108. Troyanov, M.: On the moduli space of singular Euclidean surfaces. In: Handbook of Teichmüller theory. Vol. I, IRMA Lect. Math. Theor. Phys., vol. 11, pp. 507–540. Eur. Math. Soc., Zürich (2007). https://doi.org/10.4171/029-1/13

  109. Troyanov, M.: Les surfaces à courbure intégrale bornée au sens d’Alexandrov. In: Géométrie discrète, algorithmique, différentielle et arithmétique, pp. 1–18. Société Mathématique de France (2009)

    Google Scholar 

  110. Tsuji, M.: Potential theory in modern function theory. Chelsea Publishing Co., New York (1975). Reprinting of the 1959 original

    Google Scholar 

  111. Wintner, A.: On the local role of the theory of the logarithmic potential in differential geometry. Amer. J. Math. 75, 679–690 (1953). https://doi.org/10.2307/2372542

  112. Zalgaller, V.A.: Curves on a surface near a point of the spike type. Trudy Mat. Inst. Steklov. 76, 64–66 (1965). English translation in [10]

    Google Scholar 

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Acknowledgements

The author is very grateful to Alexander Lytchak for his comments, which significantly enhanced the interest and the presentation of the present text.

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    François Fillastre

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Fillastre, F. (2023). An Introduction to Reshetnyak’s Theory of Subharmonic Distances. In: Fillastre, F., Slutskiy, D. (eds) Reshetnyak's Theory of Subharmonic Metrics. Springer, Cham. https://doi.org/10.1007/978-3-031-24255-7_4

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