The Naito--Okuda--Ouyang Biconcave Red Blood Cell Profile as a Finite-Energy Unbranched Weak Immersion with Point-Force Residues
Abstract
The axisymmetric profile sinψ(ρ)=c0ρlog(ρ/ρB) introduced by Naito, Okuda, and Ouyang is an exact solution of the unconstrained Helfrich shape equation for every ρ>0, but its mean curvature diverges logarithmically at the dimple center. A covariant local analysis shows that in Cartesian graph coordinates the immersion is nondegenerate and belongs to W2,p for every finite p; it is C1,α for every α<1, but neither C1,1 nor C2. The conformal factor has a finite nonzero limit, so the point has branch multiplicity one and is not a branch point in the sense used in weak-immersion compactness theory. The Helfrich energy is finite, whereas the complete Euler--Lagrange operator carries the distributional residue ECH=4πc0δp under the convention specified below. Hence the profile is a classical solution on the punctured surface and a finite-energy weak immersion, but it is not a free weak critical point of the unforced Canham--Helfrich functional. Stationarity is restored after adding the corresponding point-force potential or under a pinned-point constraint. Area and volume multipliers cannot cancel the Dirac residue. The apparent sign discrepancy between the shape equation used in the original papers and the modern convention is explained by the opposite definition of mean curvature. These results delimit precisely what can and cannot be inferred from modern existence and regularity theorems for Canham--Helfrich minimizers.
Related
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- Residues of weighted dynamical Ihara zeta functions on finite regular graphs
- Clifford-weighted residues, partial spectral curvature, and torsion
- Integer Sampling and the Identifiability of Mellin Residues
- Havel--Hakimi Residues of Common-Divisor Graphs: Complete Asymptotics and Prime-Counting Structure
BibTeX
@misc{wu2026naitookudaouyang,
title = {{The Naito--Okuda--Ouyang Biconcave Red Blood Cell Profile as a Finite-Energy Unbranched Weak Immersion with Point-Force Residues}},
author = {Hao Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28278},
eprint = {2609.28278},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28278},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Naito--Okuda--Ouyang Biconcave Red Blood Cell Profile as a Finite-Energy Unbranched Weak Immersion with Point-Force Residues
AU - Hao Wu
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28278
UR - https://arxiv.org/abs/2609.28278
AB - The axisymmetric profile sinψ(ρ)=c_0ρlog(ρ/ρ_B) introduced by Naito, Okuda, and Ouyang is an exact solution of the unconstrained Helfrich shape equation for every ρ>0, but its mean curvature diverges logarithmically at the dimple center. A covariant local analysis shows that in Cartesian graph coordinates the immersion is nondegenerate and belongs to W^2,p for every finite p; it is C^1,α for every α<1, but neither C^1,1 nor C^2. The conformal factor has a finite nonzero limit, so the point has branch multiplicity one and is not a branch point in the sense used in weak-immersion compactness theory. The Helfrich energy is finite, whereas the complete Euler--Lagrange operator carries the distributional residue E_CH=4πc_0δ_p under the convention specified below. Hence the profile is a classical solution on the punctured surface and a finite-energy weak immersion, but it is not a free weak critical point of the unforced Canham--Helfrich functional. Stationarity is restored after adding the corresponding point-force potential or under a pinned-point constraint. Area and volume multipliers cannot cancel the Dirac residue. The apparent sign discrepancy between the shape equation used in the original papers and the modern convention is explained by the opposite definition of mean curvature. These results delimit precisely what can and cannot be inferred from modern existence and regularity theorems for Canham--Helfrich minimizers.
ER -