Singular conformally invariant trilinear forms, I Multiplicity one results

A normalized holomorphic family (depending on $\boldsymbol \lambda \in \mathbb C^3$) of conformally invariant trilinear forms on the sphere is studied. Its zero set $Z$ is described. For $\boldsymbol \lambda\notin Z$, the multiplicity of the space of conformally invariant trilinear forms is shown to be 1, except perhaps for a denumerable subset.