曲率受控变化下的Cartan-Hadamard等周不等式
The Cartan-Hadamard isoperimetric inequality under controlled variation of curvature
- University of Wollongong（伍伦贡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在截面曲率受控变化条件下证明了任意维度的Cartan-Hadamard猜想，通过控制商极小化子的凸包并比较度量畸变与双曲等周轮廓盈余，给出关键证明步骤。
AI中文摘要：
我们证明了在所有维度下，若截面曲率相对于参考曲率半径被夹紧，则Cartan-Hadamard猜想（尖锐的欧几里得等周不等式）成立。参考曲率半径可以变化，该条件包含曲率半径趋于无穷大、流形同时具有无界（负）且渐近消失的截面曲率的例子。证明的关键步骤是在其平均曲率尺度上控制商极小化子的整个凸包，并将度量畸变与同一尺度下归一化双曲等周轮廓的盈余进行比较。
英文摘要：
We prove the Cartan-Hadamard conjecture (the sharp Euclidean isoperimetric inequality) in every dimension, assuming that the sectional curvature is pinched relative to a reference radius of curvature. The reference radius of curvature may vary, and the condition includes examples where the radius of curvature blows up, and the manifold has both unbounded (negative) and asymptotically vanishing sectional curvature. The key step in the proof is to control the entire convex hull of a quotient minimiser at its mean-curvature scale and compare the metric distortion with the surplus in the normalised hyperbolic isoperimetric profile at that same scale.
