The Schottky problem for hyperelliptic real curves

The real Torelli mapping, from the moduli space of real curves of genus g to the moduli space of g-dimensional real principally polarized abelian varieties, sends a real curve into its real Jacobian. Tile real Schottky problem is to describe its image. The results contained in the present paper concern hyperelliptic real curves and in particular real curves:of genus 2. We exhibit also some counterexamples for the non-hyperelliptic case.