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 Gårding-Dirichlet operator, such prescribed curvature problems are always solvable.

💡 Research Summary

The paper studies fully nonlinear prescribed curvature equations on closed Riemannian manifolds of dimension n ≥ 3 that have negative curvature. The central geometric object is the modified Schouten tensor \