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S-curves in polynomial external fields. (English) Zbl 1314.31006

Consider a max-min potential problem with an external field \(\varphi= \mathrm{Re\,} V\) with \(V\) a polynomial of degree \(N\geq 2\): \[ \max_\Gamma \min_{\mathrm{supp} \mu\subset\Gamma \atop \mu(\mathbb{C})=1} \left[\iint\log\frac{1}{|x-y|}d\mu(x)d\mu(y)+\int \varphi(x)d\mu(x)\right]. \] The \(\Gamma\)’s are contours and the minimal Borel probability measure \(\mu\) on \(\Gamma\) is the equilibrium measure in the external field \(\varphi\). The paper shows that this problem has a solution and that it leads to an \(S\)-curve \(\Gamma\), a notion from [A. A. Gonchar and E. A. Rakhmanov, Math. USSR, Sb. 62, No. 2, 305–348 (1989); translation from Mat. Sb., Nov. Ser. 134(176), No. 3(11), 306–352 (1987; Zbl 0663.30039)], see also [S. Kamvissis and E. A. Rakhmanov, J. Math. Phys. 46, No. 8, 083505, 24 p. (2005; Zbl 1110.81083)].
The main theorem says that an \(S\)-curve solution \(\Gamma\) can be found in a restricted set of admissible contours, the equilibrium measure is absolutely continuous and is supported on a finite union of analytic arcs that are critical trajectories of a quadratic differential. Geometrically, this means the following. The polynomial \(V\) defines \(N\) directions for which \(\varphi\to\infty\) as \(z\to\infty\) and that lie within areas of the complex plane fanning out from the central point like petals of a flower. The set of admissible contours contains curves that connect at least two points at infinity in these directions and that lie within these \(N\) petals.
The same problem was considered in [M. Bertola, Anal. Math. Phys. 1, No. 2–3, 167–211 (2011; Zbl 1259.33021)] but with a restrictive vanishing condition on \(\mu\).

MSC:

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
30C10 Polynomials and rational functions of one complex variable
60J45 Probabilistic potential theory
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