AI 中文总结
针对预定曲率方程的k-凸解，提出结合分式线性变换与双曲多项式对数凹性的新方法，通过强制性二次型估计，在3≤k<n<2k范围内建立了闭严格星形k-凸超曲面的全局曲率界。
AI 中文摘要
建立预定曲率方程的k-凸解的全局曲率估计是fully nonlinear偏微分方程和几何分析中的一个长期问题。我们开发了一种新方法，将分式线性变换与双曲多项式的对数凹性相结合。关键步骤是对极大值原理论证中三阶项产生的二次型进行强制性估计。结合一个互补的凹性不等式，该估计为满足σ_k(κ)=f(X,ν)>0的R^{n+1}中闭的、严格星形的k-凸超曲面在3≤k<n<2k范围内提供了全局曲率界。
英文摘要
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 $3\leq k<n<2k$.
Comments20 pages
