The existence of $(\mathbf{p},k)$-convex hypersurfaces for a class of Hessian quotient type curvature equations

This article investigates the existence of closed, star-shaped hypersurfaces for a class of Hessian quotient type curvature equations, in which the operator $\frac{{\sigma }_k}{{\sigma }_l}(\mathrm{\Lambda })$ arising in these equations can be viewed as a generalization of the classical Hessian quotient operator. By combining a priori estimates with the continuity method, we establish the existence and uniqueness of $(\mathbf{p},k)$-convex hypersurfaces for both nonhomogeneous and homogeneous equations of this type. Furthermore, by exploiting the recently discovered ``inverse convexity’’ property of the operator $\frac{{\sigma }_k}{{\sigma }_l}(\mathrm{\Lambda })$, we prove a constant rank theorem and thereby obtain the existence and uniqueness of strictly convex solutions to these curvature equations.