等周不等式与Cartan-Hadamard流形中的常平均曲率超曲面
The isoperimetric inequality and CMC hypersurfaces in Cartan-Hadamard manifolds
查看机构详情
- University of Science and Technology of China（中国科学技术大学）
- Georgia Institute of Technology（佐治亚理工学院）
- Fujian Normal University（福建师范大学）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
本文通过常平均曲率超曲面的尖锐不等式及边界点对积分和Jacobi场估计，证明了$3\leq n\leq9$维Cartan-Hadamard流形中的欧几里得等周不等式，确立了相应维度的Cartan-Hadamard猜想。
中文摘要 AI 辅助
我们证明了在截面曲率非正的完备单连通黎曼$n$-流形（$3\leq n\leq9$）中，对任意区域，尖锐的欧几里得等周不等式成立，从而在这些维度上确立了Cartan-Hadamard猜想。主要步骤是证明一个关于常平均曲率超曲面的尖锐不等式，该不等式通过对边界点对的积分以及沿边界测地弦的Jacobi场的估计来证明。这些积分中的权重依赖于弦的长度及其与边界的夹角，并且逐维度选取，带有Green函数极点。该不等式对于限制在测地球内的等周区域的边界仍然成立，这些边界的平均曲率仅在自由部分为常数，并且可能具有奇点。Kleiner和Ghomi-Spruck的等周轮廓论证完成了证明。
英文摘要
We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $n$-manifolds of nonpositive sectional curvature, $3\leq n\leq9$, which establishes the Cartan-Hadamard conjecture in these dimensions. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved via integrals over pairs of boundary points, together with estimates for Jacobi fields along geodesic chords of the boundary. The weights in these integrals depend on the length of the chord and its angles with the boundary, and are chosen dimension by dimension, with a Green function pole. The inequality persists for boundaries of isoperimetric regions trapped in geodesic balls, whose mean curvature is constant only on the free part and which may have singularities. The isoperimetric-profile argument of Kleiner and Ghomi-Spruck completes the proof.
