Odd branching obstructs multiplicity-one integral current structures
Abstract
Giuliano Basso asked whether every n-dimensional integral current space admits an integral current structure with the same characteristic set and multiplicity one almost everywhere. We give a negative answer already in dimension one. The obstruction is a parity phenomenon at odd-valence branch points. More precisely, if a one-dimensional integral current has unit multiplicity on the arms incident to an isolated vertex v, then the coefficient of its boundary at v has the same parity as °(v). Hence infinitely many odd-valence vertices force infinite boundary mass for every full-support unit-multiplicity current. We construct a compact geodesic, doubling, $1$-Ahlfors regular graph-like continuum G of finite length and a boundaryless current T∈I1(G) with set(T)=G, while no S∈I1(G) with set(S)=G can have multiplicity one Hh1-almost everywhere. We classify all full-support integral cycles on G. In particular, the least possible essential maximal multiplicity is exactly two, and the minimum mass of such a cycle is 3/2. For finite metric graphs we identify a complementary sharp defect: the minimum boundary mass among unit-multiplicity currents equals the number of odd-degree vertices. Finally, by taking products with flat tori and using the precise Ambrosio--Kirchheim slice representation, we obtain boundaryless compact geodesic Ahlfors-regular counterexamples in every dimension.
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BibTeX
@misc{zhong2026branching,
title = {{Odd branching obstructs multiplicity-one integral current structures}},
author = {Deguang Zhong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14515},
eprint = {2609.14515},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14515},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Odd branching obstructs multiplicity-one integral current structures
AU - Deguang Zhong
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14515
UR - https://arxiv.org/abs/2609.14515
AB - Giuliano Basso asked whether every n-dimensional integral current space admits an integral current structure with the same characteristic set and multiplicity one almost everywhere. We give a negative answer already in dimension one. The obstruction is a parity phenomenon at odd-valence branch points. More precisely, if a one-dimensional integral current has unit multiplicity on the arms incident to an isolated vertex v, then the coefficient of its boundary at v has the same parity as °(v). Hence infinitely many odd-valence vertices force infinite boundary mass for every full-support unit-multiplicity current. We construct a compact geodesic, doubling, $1$-Ahlfors regular graph-like continuum G of finite length and a boundaryless current T∈I_1(G) with set(T)=G, while no S∈I_1(G) with set(S)=G can have multiplicity one Hh^1-almost everywhere. We classify all full-support integral cycles on G. In particular, the least possible essential maximal multiplicity is exactly two, and the minimum mass of such a cycle is 3/2. For finite metric graphs we identify a complementary sharp defect: the minimum boundary mass among unit-multiplicity currents equals the number of odd-degree vertices. Finally, by taking products with flat tori and using the precise Ambrosio--Kirchheim slice representation, we obtain boundaryless compact geodesic Ahlfors-regular counterexamples in every dimension.
ER -