On the curvature of the real amoeba

Abstract

For a real smooth algebraic curve $A \subset (\mathhbb{C}^*)^2$, the amoeba $\mathcal{A}\subset {\mathbb{R}}^2$ is the image of $A$ under the map Log : $(x,y)\mapsto (\log |x|,\log |y|)$. We describe an universal bound for the total curvature of the real amoeba ${\mathcal{A}}_{\mathbb{R}A}$ and we prove that this bound is reached if and only if the curve $A$ is a simple Harnack curve in the sense of Mikhalkin.