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In mathematics, k-Hessian equations are partial differential equations based on the Hessian matrix. More specifically, a Hessian equation is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial differential equation. It can be written as S k = f {\displaystyle {\cal {S}}_{k}=f} , where 1 ⩽ k ⩽ n {\displaystyle 1\leqslant k\leqslant n} , S k = σ k ] {\displaystyle {\cal {S}}_{k}=\sigma _{k}]} , and λ = {\displaystyle \lambda =} , are the eigenvalues of the Hessian matrix D 2 u = 1 ≤ i , j ≤ n {\displaystyle {\cal {D}}^{2}u=_{1\leq i,j\leq n}} and σ k = ∑ i 1 < ⋯ < i k λ i 1 ⋯ λ i k {\displaystyle \sigma _{k}=\sum _{i_{1}<\cdots Much like differential equations often study the actions of differential operators , Hessian equations can be understood as simply eigenvalue equations acted upon by the Hessian differential operator. Special cases include the Monge–Ampère equation and Poisson's equation. The 2−hessian operator also appears in conformal mapping problems. In fact, the 2−hessian equation is unfamiliar outside Riemannian geometry and elliptic regularity theory, that is closely related to the scalar curvature operator, which provides an intrinsic curvature for a three-dimensional manifold. These equations are of interest in geometric PDEs and differential geometry.