双曲空间中退化P-Christoffel-Minkowski问题研究
Degenerate P-Christoffel-Minkowski Problems in Hyperbolic Space
杨彬彬 1徐露2
作者信息
- 1. 湖南大学数学学院,长沙 410082
- 2. 湖南大学数学学院,长沙 410082
- 折叠
摘要
本文将退化p-Christoffel-Minkowski问题推广至双曲空间,借鉴欧氏空间中的方法研究极限球面退化p-Christoffel-Minkowski 问题。在凸体光滑性假设下,将该问题转化为单位球面上的曲率方程；首先对右端函数施加适当正则性条件,得到不依赖其下界的低阶估计；进而建立一致\(C^{1,1}\)估计,借助关键引理处理协变导数在交换顺序时不满足Codazzi条件所产生的“坏项”；在非退化情形下给出正则性估计后,通过光滑逼近退化情形,结合一致估计与紧性定理取极限,最终完成退化情形的证明。
Abstract
We extend the degenerate \(p\)-Christoffel–Minkowski problem to hyperbolic space and study its horospherically degenerate case by adapting methods from Euclidean space. Under smooth convexity assumptions, the problem is reduced to a curvature equation on the unit sphere. Lower-order estimates independent of the lower bound are derived under suitable regularity conditions on the right-hand side function. Uniform \(C^{1,1}\) estimates are established, with a key lemma handling bad terms arising from non-Codazzi tensors when commuting covariant derivatives. After obtaining regularity estimates in the non-degenerate case, we prove the degenerate case via smooth approximation, uniform estimates and compactness.关键词
基础数学/p-Christoffel-Minkowski 问题/极限球面/退化/Key words
Basic Mathematics/\(p\)-Christoffel-Minkowski problem/horosphere/degeneracy引用本文复制引用
杨彬彬,徐露.双曲空间中退化P-Christoffel-Minkowski问题研究[EB/OL].(2026-04-28)[2026-04-29].http://www.paper.edu.cn/releasepaper/content/202604-205.学科分类
数学
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