TITLE: Visualizing holomorphic differentials on a compact Riemann surface?
QUESTION [9 upvotes]: It is a classical result that the vector space of holomorphic differentials on a compact Riemann surface of genus $g$ has dimension $g$.  I am wondering if there is a way of visualizing this wonderful result.
I should perhaps say a bit more about what I mean by "visualizing."  I found another MO question on visualizing the Riemann–Roch theorem, and what I am asking for here could be viewed as the simplest case of Riemann–Roch.  There are some interesting answers to that question, but I am hoping for something more literally visual, along the lines of Dan Piponi's note On the visualisation of differential forms and/or the paper On the geometry, flows and visualization of singular complex analytic vector fields on Riemann surfaces by Alvarez-Parrilla et al.  The ultimate dream would be some kind of animation that allows one to get some intuitive feeling for what holomorphic (or harmonic, if that is easier) differentials are constrained to be like, and why adding an extra hole adds an extra degree of freedom.

REPLY [6 votes]: This is not really an answer, it is a comment with images.
The paper "Quadrilateral Mesh Generation II : Meromorphic Quartic Differentials and Abel-Jacobi Condition" (arXiv 2019) contains (imo) nice exposition of theoretical background and several illuminating illustrations.

REPLY [5 votes]: There is a very nice hydrodynamic interpretation, going back to Riemann and Klein, and popularized by Courant (Hurwitz and  Courant, Function theory). For those who do not read German, there is a modern exposition in French:
E. Ghys and D. Smai,
Six lecons autour des surfaces de Riemann
Edit. Klein's book is "On Riemann's theory of algebraic functions", the only one of the three which has been translated into English.