Yamabe-Type Equations Under Tensorial Perturbations of the Conformal Class on Closed Riemannian Manifolds

We establish a perturbative stability result for the Yamabe problem under genuinely non-conformal tensorial deformations on closed Riemannian manifolds \( (M^n,g) \), \( n \ge 3 \), incorporating small symmetric perturbations \( T \in C^{2,\alpha}(S^2 T^* M) \) beyond classical conformal rescalings. By introducing a tensorial correction \( E[T,\nabla T] \) in the scalar curvature functional, we define a perturbed variational problem whose critical points satisfy the modified Euler--Lagrange equation \( -a \Delta_g \Omega + (R_g + E[T,\nabla T]) \Omega = \lambda \, \Omega^p, \quad p = \frac{n+2}{n-2} \). Using precise linearization of Ricci and scalar curvatures, we derive quantitative estimates for the trace-free Ricci contributions and prove a rigidity theorem on Einstein backgrounds: sufficiently small perturbations T cannot generate a nontrivial trace-free Ricci component. Moreover, we establish perturbative existence and uniqueness of solutions \( \Omega_T \) in the perturbed conformal class \( \mathcal{C}(g,T) \), with explicit control \( \|\Omega_T - \Omega_0\|_{C^{2,\alpha}} \le C \, \|T\|_{C^{2,\alpha}} \). Our analysis provides a rigorous framework connecting classical Yamabe theory to tensorial deformations, yielding sharp stability estimates, bounds on linear and higher-order contributions, and explicit conditions under which Einstein metrics remain locally rigid. These results form a foundation for future investigations on higher-order curvature operators, large perturbations, and bifurcation phenomena in generalized Yamabe-type problems.