Second fundamental form and higher Gaussian maps

In this paper we show a relation between higher even Gaussian maps of the canonical bundle on a smooth projective curve of genus $g \geq 4$ and the second fundamental form of the Torelli map. This generalises a result obtained by Colombo, Pirola and Tortora on the second Gaussian map and the second fundamental form. As a consequence, we prove that for any non-hyperelliptic curve, the Gaussian map $\mu_{6g-6}$ is injective, hence all even Gaussian maps $\mu_{2k}$ are identically zero for all $k >3g-3$. We also give an estimate for the rank of $\mu_{2k}$ for $g-1 \leq k \leq 3g-3.$