Global Curvature Estimates for k-Convex Hypersurfaces

Establishing global curvature estimates for $k$-convex solutions of prescribed curvature equations is a longstanding problem in fully nonlinear partial differential equations and geometric analysis. We develop a new approach that combines fractional-linear transformations with the logarithmic concavity of hyperbolic polynomials. The key step is a coercive estimate for the quadratic forms arising from the third-order terms in the maximum-principle argument. Together with a complementary concavity inequality, this estimate yields global curvature bounds for closed, strictly star-shaped $k$-convex hypersurfaces in ${\mathbb{R}}^{n+1}$ satisfying ${\sigma }_k(\kappa )=f(X,\nu )>0$, throughout the range $3\le k<n<2k$.