PreprintarXiv licencearXiv
The Cartan-Hadamard conjecture in dimension five
Abstract
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 $3$.
Related
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- A proof of the Cartan-Hadamard conjecture for small volumes under a Ricci curvature lower bound
- Petty's equilateral-set conjecture in dimension five
- Isoperimetric and total curvature inequalities in Cartan-Hadamard manifolds with nullity
- Almgren frequency and Agmon convexity on rotationally symmetric Cartan-Hadamard manifolds
BibTeX
@misc{chen2026cartanhadamard,
title = {{The Cartan-Hadamard conjecture in dimension five}},
author = {Shibing Chen and Mohammad Ghomi and Peng Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11005},
eprint = {2609.11005},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11005},
}
RIS
TY - UNPB
TI - The Cartan-Hadamard conjecture in dimension five
AU - Shibing Chen
AU - Mohammad Ghomi
AU - Peng Wang
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11005
UR - https://arxiv.org/abs/2609.11005
AB - 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 $3$.
ER -