{"id":"openai-math/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity","pdf":"https://github.com/openai/math/blob/main/preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026","tags":["differential-geometry","isoperimetric-inequalities","comparison-geometry"],"score":0,"title":"Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity","domain":"Differential geometry","family":"337","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"We resolve the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most κ ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature κ . For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":false,"family_title":"Sharp Cartan–Hadamard isoperimetry and rigidity"}