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

Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces