  We study global attractors $\mathcal{A}=\mathcal{A}_f$ of semiflows generated
by semilinear partial parabolic differential equations of the form $u_t =
u_{xx} + f(x,u,u_x), 0<x<1$, satisfying Neumann boundary conditions. The
equilibria $v\in\mathcal{E}\subset\mathcal{A}$ of the semiflow are the
stationary solutions of the PDE, hence they are solutions of the corresponding
second order ODE boundary value problem. Assuming hyperbolicity of all
equilibria, the dynamic decomposition of $\mathcal{A}$ into unstable manifolds
of equilibria provides a geometric and topological characterization of Sturm
global attractors $\mathcal{A}$ as finite regular signed CW-complexes, the
Sturm complexes, with cells given by the unstable manifolds of equilibria.
Concurrently, the permutation $\sigma=\sigma_f$ derived from the ODE boundary
value problem by ordering the equilibria according to their values at the
boundaries $x=0,1$, respectively, completely determines the Sturm global
attractor $\mathcal{A}$. Equivalently, we use a planar curve, the meander
$\mathcal{M}=\mathcal{M}_f$, associated to the the ODE boundary value problem
by shooting.
  The main objective of this paper is to derive a minimax property which
identifies the equilibria on the cell boundary of $\mathcal{O}$ which are
closest or most distant from $\mathcal{O}$ at the boundaries $x=0,1$, directly
from the permutation $\sigma$, the Sturm permutation, or equivalently from the
meander $\mathcal{M}$, the Sturm meander, based on the Sturm nodal properties
of the solutions of the ODE boundary value problem. We emphasize the local
aspect of this result by applying it to an example for which the identification
of the equilibria in the cell boundary of $\mathcal{O}$ is obtained from the
knowledge of only a section of the Sturm meander $\mathcal{M}$.
