$N(k)$-contact metric as $\ast$-conformal Ricci soliton

The aim of this paper is characterize a class of contact metric manifolds admitting $\ast$-conformal Ricci soliton. It is shown that if a $(2n + 1)$-dimensional$N(k)$-contactmetricmanifold$M$admits$\ast$-conformal Ricci soliton or $\ast$-conformal gradient Ricci soliton, then the manifold M is $\ast$-Ricci at and locally isometric to the Riemannian of a flat $(n + 1)$-dimensionalmanifoldandan$n$-dimensional manifold of constant curvature 4 for $n>1$ and flat for $n=1$. Further, for the first case, the soliton vector field is conformal and for the $\ast$-gradient case, the potential function $f$ is either harmonic or satisfy a Poisson equation. Finally, an example is presented to support the results.