We investigate a one-parameter deformation of the Ricci flow arising from the Lagrangian \(L = f(R) = R^{1 + \varepsilon }\) where \(\varepsilon\) is a small real parameter. Varying this action with respect to the metric, yields a fourth-order geometric flow, that reduces to the Ricci flow as \(\varepsilon \to 0\) . The corresponding evolution equation is: \(\partial_{t} g_{ij} = - 2{\mathcal{E}}_{ij} = - 2\left( {1 + \varepsilon } \right)R^{\varepsilon } R_{ij} + g_{ij} R^{1 + \varepsilon } + 2\left( {1 + \varepsilon } \right)\left( {\nabla_{i} \nabla_{j} - g_{ij} {\square }} \right)R^{\varepsilon }\) . We derive the evolution of the scalar curvature, its small—expansion, and the associated fourth-order corrections to the standard Ricci flow. Linear analysis shows that for \(\varepsilon > 0\) , the flow exhibits higher-order diffusion through bi-Laplacian-type terms, modifying the stability of Einstein metrics. Soliton (self-similar) solutions satisfy a generalized equation involving the Hessian of a potential \(f\) and the curvature scalar \(R^{\varepsilon }\) . For constant-curvature (Einstein) metrics, we find explicit solitons with scaling parameter \(\sigma = - \frac{1 + n}{\varepsilon } - \frac{1}{2}R^{1 + \varepsilon } ,\) whose sign determines whether the soliton is shrinking, steady, or expanding. We also present a generalized formulation of the Ricci flow derived from a deformed Perelman functional: \({\mathbf{\mathcal{F}}}_{\varepsilon } [f,g_{ij} ]: = \int\limits_{M} {e^{ - f} \left( {\left| {\nabla f} \right|^{2} + R} \right)^{1 + \varepsilon } dV} ,\) where \(\varepsilon\) is a real deformation parameter. By varying this functional with respect to the metric \(g_{ij}\) and scalar field \(f\) we obtain a modified Ricci flow system in which the classical Perelman equations are rescaled by the curvature–gradient factor \((\left| {\nabla f} \right|^{2} + R)^{\varepsilon }\) and supplemented by additional higher-order coupling terms involving derivatives of \(L = \left| {\nabla f} \right|^{2} + R\) . The resulting evolution equations preserve the parabolic nature of the Ricci flow while introducing nontrivial corrections that affect entropy and monotonicity properties. A small- \(\varepsilon\) expansion yields explicit first-order corrections of the form \(L\ln L\) , revealing the mechanism through which Perelman’s entropy monotonicity is perturbed. This deformation provides a controlled one-parameter extension of Ricci flow dynamics, offering a framework for exploring generalized geometric flows, entropy production, and stability analysis in Riemannian geometry. We also study a generalization of the Ricci flow driven by a combined functional incorporating the \(L^{1 + \varepsilon (t)}\) norm of the curvature tensor, where \(\varepsilon (t)\) is a time-dependent exponent. This formulation leads to a nonautonomous geometric flow in which the curvature regions are weighted dynamically, and introduces a mixed 2nd- and 4th-order structure in the evolution equations. Of particular interest are soliton solutions whose existence and stability are strongly influenced by the temporal behavior of the Lagrangian exponent \(1 + \varepsilon (t)\) . The time-dependence affects both the gradient structure of the functional and the balance between Ricci diffusion and curvature concentration, thereby playing a central role in determining whether self-similar or perturbative solitons can arise. Our analysis highlights the sensitivity of soliton formation to variations in \(\varepsilon (t)\) , offering a framework to study new families of generalized solitons and their geometric properties. Two conjectures have been also proposed.
