Locally convex hypersurfaces of constant curvature with boundary

for some constant K, where κ[M ] = (κ1, . . . , κn) denotes the principal curvatures of M . Important examples include the classical Plateau problem for minimal or constant mean curvature surfaces and the corresponding problem for Gauss curvature, which was treated recently by the authors [12] and independently by Trudinger-Wang [24]. In this paper as in our previous work [12], we are concerned with locally convex hypersurfaces. Accordingly, the function f is assumed to be defined in the convex cone Γn ≡ { λ ∈ R : each component λi > 0 } in R and satisfy the fundamental structure conditions: