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arXiv · 2609.15858

Global Curvature Estimates for k-Convex Hypersurfaces

Fengrui Yang
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Abstract

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 $σ_k(κ)=f(X,ν)>0$, throughout the range $3\leq k<n<2k$.

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Fengrui Yang. 2026-09-14. Global Curvature Estimates for k-Convex Hypersurfaces. https://arxiv.org/abs/2609.15858

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