Dirichlet energy and focusing NLS condensates of minimal intensity

We consider the family of (poly)continua $\text{K}$ in the upper half-plane $\mathbb{H}$ that contain a preassigned finite {\it anchor} set $E\in \mathbb{H}$. For a given harmonic external field we define a Dirichlet energy functional $\mathcal{I}(\mathcal{K})$ and show that within each connectivity class'' of the family, there exists a minimizing compact $\mathcal K^*$ consisting of critical trajectories of a quadratic differential. In many cases this quadratic differential coincides with the square of the real normalized quasimomentum differential ${\rm d} {\bf p}$ associated with the finite gap solutions of the focusing Nonlinear Schrödinger equation (fNLS) defined by a hyperelliptic Riemann surface $\mathfrak R$ branched at the points $E\cup\bar E$. The motivation for this work lies in the problem of soliton condensate of least average intensity such that a given anchor set $E$ belongs to the poly-continuum $\mathcal K$. An fNLS soliton condensate is defined by a compact $\mathcal K\subset{\mathbb H} $ (its spectral support) whereas the average intensity of the condensate is proportional to $\mathcal I(\mathcal K)$. We prove that the spectral support $\mathcal K^*$ provides the fNLS soliton condensate of the least average intensity within a given connectivity class’'.