In this article, we derived an equality for $\mathcal{CR}$-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a $\mathcal{CR}$-warped product submanifolds in K$\ddot{a}$hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classification of $\mathcal{CR}$-warped product submanifolds in the K$\ddot{a}$hler manifold by using the Euler–Lagrange equation, Dirichlet energy and Hamiltonian is given. We also derive some characterizations of Einstein warped product manifolds under the impact of Ricci Curvature and Divergence of Hessian tensor.