  A distance may be defined on a domain of the plane, by considering a suitable
set of admissible curves, and a function $\lambda$ such that its square root is
integrable along any admissible curve. More precisely, to integrate the square
root of $\lambda$ gives a notion of length of the curve, and taking the infimum
of these lengths between two points endows the domain with a pseudo-distance.
If $\lambda$ is $C^\infty$, one obtains a Riemannian distance. Other basic
famous examples are flat metrics with conical singularities. What should be the
good regularity of $\lambda$ to develop a theory of "Riemannian geometry with
low regularity" on surfaces, containing the Riemannian case and the case of
flat metrics? This has been answered in a series of articles by \res in the
$60'$s: the logarithm of $\lambda$ is a difference of subharmonic functions.
  These distances come with a natural notion of curvature measure,and a central
result in the theory is that, roughly speaking, the converge of the measures
implies the convergence of the distances. On a given domain, the set of
subharmonic distances is exactly the set of the limits (for the uniform
convergence) of sequences of Riemannian distances with uniformly bounded total
variation of the curvature measure.
  In the same period, A. Huber proved that a distance preserving mapping
between domains with subharmonic distances is a conformal mapping, giving a
bridge with the theory of Riemann surfaces. Also, using his convergence
theorem, \res proved that subharmonic distances and two-dimensional manifolds
of bounded curvatures, in the sense of A.D. Alexandrov, are the same thing,
providing an analytic approaches to a classical objects of the theory of metric
geometry of surfaces.
