Fully nonlinear prescribed curvature problems on closed manifolds with negative curvature

In this manuscript, we investigate fully nonlinear prescribed curvature problems for the modified Schouten tensor on closed Riemannian manifolds with negative curvature. We prove that whenever the corresponding concave elliptic operator satisfies a structural Condition $T$, which encompasses all $O(n)$-invariant G\r{a}rding-Dirichlet operator, such prescribed curvature problems are always solvable.