Amoebas of maximal area

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To any algebraic curve A in a complex 2-torus ()² one may associate a closed infinite region in a real plane called the amoeba of A. The amoebas of different curves of the same degree come in different shapes and sizes. All amoebas in have finite area and, furthermore, there is an upper bound on the area in terms of the degree of the curve. The subject of this paper is the curves in a complex 2-torus whose amoebas are of the maximal area. We show that up to multiplication by a constant such curves are defined over real numbers and, furthermore, that their real loci are isotopic to so-called Harnack curves.