
## 4. $S$-curves

1. ### $S$-curves for complex potentials in the scalar case

#### Problem 4.1.

[A. Kuijlaars] Analysis of existence and properties of $S$-curves in the scalar case when the potential is $\phi(z)=\textrm{Re}\, z^p$, for $p\geq 3$, and we take $\tau$ as the set of continua that join neighboring valleys of $\phi(z)$ (where $\phi(z)\to-\infty$) in the complex plane. In particular, prove that the support of the equilibrium measure on a curve with the $S$-property is one-cut case, or give bounds on the number of cuts depending on $p$.
• ### Numerical computation of $S$-curves

#### Problem 4.2.

[M. Bertola] A general question about numerical methods for vector equilibrium problems and computation of corresponding $S$-curves. More in particular, how to produce critical trajectories in a reliable way, given the spectral curve of the problem?
• ### $S$-curves with piecewise harmonic external field

#### Problem 4.3.

[E. Rakhmanov] Consider the external potential $$\phi(z)=\begin{cases} -k_1\, \textrm{Re}\, z, & \textrm{Re}\, z>0,\\ k_2\, \textrm{Re}\, z, & \textrm{Re}\, z<0, \end{cases}$$ where $k_1,k_2>0$. Is there a continuum with the $S$-property connecting the points $z_1=-\varepsilon_1+i$ and $z_2=\varepsilon_2-i$ in the complex plane, for small positive $\varepsilon_1$ and $\varepsilon_2$? Also, consider the max-min problem in this setting. Does the curve obtained have a portion along the imaginary axis?
• ### Green’s potential

#### Problem 4.4.

[A. Martínez-Finkelshtein] Existence theory for $S$-curves in a general sense, if the logarithmic potential is replaced by Green’s potential.

Cite this as: AimPL: Vector equilibrium problems and random matrix models, available at http://aimpl.org/vectorequilib.