发表机构
University of Science and Technology of China; Georgia Institute of Technology; Fujian Normal University（中国科学技术大学; 佐治亚理工学院; 福建师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了五维完备单连通非正截面曲率流形上的欧几里得等周不等式，确立了Cartan-Hadamard猜想，方法基于边界点对积分与Jacobi场估计，并推广至三维。
AI 中文摘要
我们证明了尖锐的欧几里得等周不等式对于完备单连通、截面曲率非正的黎曼$5$维流形中的区域成立，从而在该维度上确立了Cartan-Hadamard猜想。主要步骤是一个关于常平均曲率超曲面的尖锐不等式，通过边界点对的积分证明，遵循Banchoff-Pohl的精神，并结合沿测地弦的Jacobi场的估计。该不等式对于测地球中等周区域的边界仍然成立，其平均曲率仅在自由部分为常数。继Kleiner之后的等周剖面论证完成了证明。我们的方法也在$3$维中给出了新的证明。
英文摘要
We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $5$-manifolds of nonpositive sectional curvature, which establishes the Cartan-Hadamard conjecture in that dimension. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved via integrals over pairs of boundary points, in the spirit of Banchoff-Pohl, together with an estimate for Jacobi fields along geodesic chords. The inequality persists for boundaries of isoperimetric regions in geodesic balls, whose mean curvature is constant only on the free part. An isoperimetric-profile argument, after Kleiner, completes the proof. Our method also gives a new proof in dimension $3$.
Comments19 pages
