Abstract: Lens spaces L(p,q) are a family of closed 3-manifolds indexed by two coprime integers. They can be described as quotients of the 3-sphere by free isometric actions of cyclic groups; hence they carry Riemannian metrics of (constant) positive sectional curvature. Alternatively, they can be obtained by gluing together two solid tori along their common boundary, which is called a Heegaard torus. Our main theorem is as follows: fix a metric of positive Ricci curvature on L(p,q), and denote by M(p,q) the moduli space of Heegaard tori in L(p,q) that have positive mean curvature. If q = ±1 mod p, then M(p,q) is path-connected. Otherwise it has exactly two path-components. This is work in progress, joint with Sylvain Maillot (Université de Montpellier).