Properties of level curves of a function generated by an Abel integral on three-sheeted torus

We consider some Abel integral $F$ on a three-sheeted torus $T$ over the Riemann sphere $\mathbb{C}$ which is the Riemann surface of the function $w=\sqrt[3]{(z-a_1)(z-a_2)(z-a_3)}$; here $a_k$, $1\le k\le 3$ are pairwise distinct points of $\mathbb{C}$. We describe the corresponding Abel integral on the universal covering of $T$ and the differential-geometric structure of level lines of $\mathrm{Re}\,F$ which is a single-valued harmonic function. With the help of the study of the zero level line of the function, we can give a description of so-called Nuttall decomposition of $T$ which plays an important role in the theory of Hermite–Padé diagonal approximations II.

If the points $a_k$ are the vertices of some isosceles triangle, we can also completely describe the projection of the zero level line of $\mathrm{Re}\,F$ on $\mathbb{C}$. Our technique is based on the theory of the Weierstrass elliptic functions.