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An Introduction to Reshetnyak’s Theory of Subharmonic Distances

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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.


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

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  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].


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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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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.

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