The Naito--Okuda--Ouyang Biconcave Red Blood Cell Profile as a Finite-Energy Unbranched Weak Immersion with Point-Force Residues

Abstract

The axisymmetric profile \(\sinψ(ρ)=c_0ρ\log(ρ/ρ_B)\) 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.