Gentle $m$-Calabi-Yau tilted algebras

We prove that all gentle 2-Calabi-Yau tilted algebras (over an algebraically closed field) are Jacobian, moreover their bound quiver can be obtained via block decomposition. Related families of gentle $(m+1)$-Calabi-Yau tilted algebras are the $m$-cluster-tilted algebras of type $\mathbb{A}$ and $\tilde{\mathbb{A}}$. For these algebras we prove that a module $M$ is stable Cohen-Macaulay if and only if ${\mathrm{\Omega }}^{m+1}\tau M\simeq M$.