  We investigate the logarithmic convexity of the length of the level curves
for harmonic functions on surfaces and related isoperimetric type inequalities.
The results deal with smooth surfaces, as well as with singular Alexandrov
surfaces (also called surfaces with bounded integral curvature), a class which
includes for instance surfaces with conical singularities and surfaces of
CAT(0) type. Moreover, we study the geodesic curvature of the level curves and
of the steepest descent for harmonic functions on surfaces with non-necessarily
constant Gaussian curvature $K$. Such geodesic curvature functions turn out to
satisfy certain Laplace-type equations and inequalities, from which we infer
various maximum and minimum principles.
  The results are complemented by a number of growth estimates for the
derivatives $L'$ and $L''$ of the length of the level curve function $L$, as
well as by examples illustrating the presentation.
  Our work generalizes some results due to Alessandrini, Longinetti, Talenti,
Ma--Zhang and Wang--Wang.
