Curvature Estimates and Prescribed Curvature Problems for Sum Hessian Equations on Star-Shaped Hypersurfaces

Abstract

Let $M$ be a closed star-shaped hypersurface in $\mathbb{R}^{n+1}$ and let $η=Hg-h$ be the first Newton transformation of its second fundamental form. For a fixed constant $α>0$ and $2\le k\le n-1$, we study the prescribed sum Hessian curvature equation \[ σ_k(λ(η))+ασ_{k-1}(λ(η))=f(X,ν). \] Under the natural admissibility condition \[ λ(η)\inΓ_{k-1}\cap \{λ:σ_k(λ)+ασ_{k-1}(λ)>0\}, \] we establish a global curvature estimate for closed star-shaped solutions with a general positive right-hand side depending on the position vector and the unit normal. A key point is that the sum Hessian operator is nonhomogeneous; the resulting lower-order term in the maximum-principle calculation is controlled by a partial curvature estimate. We also use the augmented-variable identity \[ σ_k(λ)+ασ_{k-1}(λ) =σ_k(λ_1,\ldots,λ_n,α) \] to recover the relevant ellipticity and concavity properties. As an application of the a priori estimates, we prove existence and uniqueness for the corresponding prescribed curvature problem under suitable spherical barrier and radial monotonicity assumptions.