4. $S$curves

$S$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 onecut 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 $z_2=\varepsilon_2i$ in the complex plane, for small positive $\varepsilon_1$ and $\varepsilon_2$? Also, consider the maxmin problem in this setting. Does the curve obtained have a portion along the imaginary axis? 
Green’s potential
Problem 4.4.
[A. MartínezFinkelshtein] 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.