Global Curvature Estimation For The Solutions Of A Class Of Hessian Type Prescribed Curvature Equations

In this paper,we mainly study the curvature estimates of the solution of prescribed curvature equation in the following forms:σ_k（κ（X）+b（X））=ψ（X,υ）,X∈M.（0.1）Where M?Rⁿ⁺¹is an n-dimensional closed hypersurface.Suppose X,υrepresent the point on surface M and the normal vector outside the unit of the point.Supposeσ_kis k-elementary symmetric function.Letκ（X）is the principal curvature at point X.b（X）is some given smooth symmetric 2-tensor.ψ（X,υ）>0 is a given smooth function.In this paper,we mainly discuss the curvature estimation of the prescribed curvature e-quation（0.1）.For generalψ^1/k（X,υ）>0,the curvature estimation of k+1convex solution of equation（0.1）is established.Namely,whenκ（X）∈Γ_n,b（X）∈Γ_k+1,the curvature of the equation（0.1）has a priori upper bound.