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Generic Uniqueness of Isoperimetric Regions in Arbitrary Dimension
Abstract
Let Mn+1 be a closed, connected smooth manifold, where n≥ 1. For each prescribed volume fraction s∈(0,1)∖{1/2}, we prove that the isoperimetric region of volume s Volg(M) is unique for a generic set of smooth Riemannian metrics g. At half volume, a generic metric has exactly two minimizers, a region E and its complement Ec. As consequences, for every m>0, uniqueness holds for a generic set of metrics g satisfying m<Volg(M), and it also holds for a generic set of pairs (g,m) with 0<m<Volg(M). The proof is variational and requires neither boundary regularity nor nondegeneracy of the constrained Jacobi operator. In particular, the results apply even when isoperimetric boundaries are singular.
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BibTeX
@misc{niu2026generic,
title = {{Generic Uniqueness of Isoperimetric Regions in Arbitrary Dimension}},
author = {Gongping Niu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20790},
eprint = {2609.20790},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20790},
}
RIS
TY - UNPB
TI - Generic Uniqueness of Isoperimetric Regions in Arbitrary Dimension
AU - Gongping Niu
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20790
UR - https://arxiv.org/abs/2609.20790
AB - Let M^n+1 be a closed, connected smooth manifold, where n≥ 1. For each prescribed volume fraction s∈(0,1)∖{1/2}, we prove that the isoperimetric region of volume s Vol_g(M) is unique for a generic set of smooth Riemannian metrics g. At half volume, a generic metric has exactly two minimizers, a region E and its complement E^c. As consequences, for every m>0, uniqueness holds for a generic set of metrics g satisfying m<Vol_g(M), and it also holds for a generic set of pairs (g,m) with 0<m<Vol_g(M). The proof is variational and requires neither boundary regularity nor nondegeneracy of the constrained Jacobi operator. In particular, the results apply even when isoperimetric boundaries are singular.
ER -