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Global Curvature Estimates for k-Convex Hypersurfaces
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 ℝn+1 satisfying σk(κ)=f(X,ν)>0, throughout the range 3≤ k<n<2k.
Related
- Curvature integral on convex hypersurfaces
- Curvature Estimates and Prescribed Curvature Problems for Sum Hessian Equations on Star-Shaped Hypersurfaces
- Interior curvature estimates for convex solutions of a prescribed curvature quotient equation
- Interior Curvature Estimates of Semi-convex Solutions for the scalar curvature equation with Lipschitz Right-Hand Sides
- Hypersurfaces with constant sectional curvature of a product of two space forms
BibTeX
@misc{yang2026global,
title = {{Global Curvature Estimates for k-Convex Hypersurfaces}},
author = {Fengrui Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15858},
eprint = {2609.15858},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15858},
}
RIS
TY - UNPB
TI - Global Curvature Estimates for k-Convex Hypersurfaces
AU - Fengrui Yang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15858
UR - https://arxiv.org/abs/2609.15858
AB - 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 ℝ^n+1 satisfying σ_k(κ)=f(X,ν)>0, throughout the range 3≤ k<n<2k.
ER -