A class of Hessian quotient equations in warped product manifolds

Given a compact Riemannian manifold M and an open interval I in R, we consider a warped product manifold (M) over bar = I x(phi)M. For a positive function f defined on (M) over bar, we obtain the existence of the (eta, k)-convex hypersurface Sigma which satisfies a Hessian quotient equation sigma(k)(lambda(eta))/sigma(l)(lambda(eta)) = f(V,nu(V)) for 0 <= l < k < n, and eta = Hg - h, the first Newton transformation of the second fundamental form h. This generalizes the result of Chen et al. (2020) and gives a simpler proof of the curvature estimate. As a corollary, we can get an (eta, k)-convex solution for f(V,nu) = < V,nu >|V|(-n-1)h(V|V|) in Rn+1 for some prescribed function h, which can be viewed as the prescribed curvature measure type problem.