Initial layer of the anti-cyclotomic $\mathbb{Z}_3$-extension of $\mathbb{Q}(\sqrt{-m})$ and capitulation phenomenon

Let $k=\mathbb{Q}(\sqrt{-m})$ be an imaginary quadratic field. We consider the properties of capitulation of the $p$-class group of $k$ in the anti-cyclotomic $\mathbb{Z}_p$-extension $k^{\rm ac}$ of $k$; for this, using a new algorithm, we determine for $p=3$ the first layer $k_1^{\rm ac}$ of $k^{\rm ac}$ over $k$ (with absolutely general pari/gpprograms), and we examine if, at least, some partial capitulation may exist in $k_1^{\rm ac}$. The answer seems obviously yes ,even when $k^{\rm ac}/k$ is totally ramified. We have conjectured that this phenomenon of capitulation is specific of all the $\mathbb{Z}_p$-extensions of $k$, distinct from the cyclotomic one, whatever $p$. We characterize a family of fields $k=\mathbb{Q}(\sqrt{-m})$ (called Normal Split cases) for which $k^{\rm ac}$ is not linearly disjoint from the Hilbert class field (Theorem 2.7). Many numerical illustrations are given with computation of the $3$-class group of $k_1^{\rm ac}$. No assumptions are made on the structure of the 3-class group of $k$, nor on the splitting of 3 in $k$ and in its mirror field $k^*=\mathbb{Q}(\sqrt{3m})$.