双曲空间中一类混合 k-Hessian 方程\\的先验估计
A Priori Estimates for a Class of Mixed k-Hessian Equations in Hyperbolic Space
摘要
$L_p$ Minkowski 问题是现代凸几何领域的前沿热点。本文在负曲率的双曲空间 $\mathbb{H}^{n+1}$ 背景下,研究了由该问题诱导的一类高阶完全非线性混合 $k$-Hessian 方程。针对双曲空间中对称 2-张量不满足经典 Codazzi 属性的几何难点,本文推导了该背景下的高阶导数交换公式。结合极值原理与广义 Newton-MacLaurin 不等式,本文成功克服了方程中非线性项的高阶增长困难,严格建立了该方程解的 $C^0$、$C^1$ 及 $C^2$ 全局闭先验界。本研究不仅完善了双曲空间中高阶几何方程的解析推导技巧,也为进一步利用连续性方法或拓扑度理论探讨该几何方程的全局平滑解提供了关键的理论支撑。
Abstract
The $L_p$ Minkowski problem is a frontier issue in the field of modern convex geometry. Focusing on the negative curvature setting of the hyperbolic space $\mathbb{H}^{n+1}$, this paper investigates a class of higher-order fully nonlinear mixed $k$-Hessian equations induced by this problem. To address the geometric difficulty that symmetric 2-tensors lack the classical Codazzi property in hyperbolic space, this study derives specific commutation formulas for higher-order derivatives. By combining the maximum principle with generalized Newton-MacLaurin inequalities, this paper successfully overcomes the difficulties caused by the higher-order growth of nonlinear terms, rigorously establishing $C^0$, $C^1$, and $C^2$ global closed a priori bounds for the solutions. This research not only enriches the analytical derivation techniques for higher-order geometric equations in hyperbolic space, but also provides critical theoretical support for further investigating the existence of global smooth solutions to this equation using the continuity method or degree theory.关键词
双曲空间/$L_p$ Minkowski 问题/混合 $k$-Hessian 方程/先验估计Key words
Hyperbolic space/$L_p$ Minkowski problem/Mixed $k$-Hessian equation/A priori estimates引用本文复制引用
黄俊龙,徐露.双曲空间中一类混合 k-Hessian 方程\\的先验估计[EB/OL].(2026-05-28)[2026-06-03].http://www.paper.edu.cn/releasepaper/content/202605-127.学科分类
数学
