  Given a connected set $\Omega_0 \subset \mathbb{R}^2$, define a sequence of
sets $(\Omega_n)_{n=0}^{\infty}$ where $\Omega_{n+1}$ is the subset of
$\Omega_n$ where the first eigenfunction of the (properly normalized) Neumann
$p-$Laplacian $ -\Delta^{(p)} \phi = \lambda_1 |\phi|^{p-2} \phi$ is positive
(or negative). For $p=1$, this is also referred to as the Ratio Cut of the
domain. We conjecture that, unless $\Omega_0$ is an isosceles right triangle,
these sets converge to the set of rectangles with eccentricity bounded by 2 in
the Gromov-Hausdorff distance as long as they have a certain distance to the
boundary $\partial \Omega_0$. We establish some aspects of this conjecture for
$p=1$ where we prove that (1) the 1-Laplacian spectral cut of domains
sufficiently close to rectangles of a given aspect ratio is a circular arc that
is closer to flat than the original domain (leading eventually to
quadrilaterals) and (2) quadrilaterals close to a rectangle of aspect ratio $2$
stay close to quadrilaterals and move closer to rectangles in a suitable
metric. We also discuss some numerical aspects and pose many open questions.

Spectral Theory