Critical bifurcations in deformable membrane necks under inhomogeneous curvature: constriction frustration vs. abscissional elongation

Catenoid necks, as minimal surfaces with zero mean curvature ( K = 0 ) , minimize bending energy and serve as geometric scaffolds for scissional membrane remodeling. We apply the Canham–Helfrich model of flexible membranes to analyze deformable spontaneous curvature ( K 0 ) , a key regulator of membrane scission events in cellular compartmentalization. To model functional membrane necking, we examine deformed catenoidal shapes with variable mean curvature ( δ K ≠ 0 ) near the minimal-energy catenoid ( K = 0 ) , which varies along either the constrictional or elongational pathways. Using the Euler–Lagrange equilibrium equations, we derive inhomogeneous catenoid solutions, revealing metastable singularities departing from the critical catenoid of the maximal area—a tipping point (TP) for scission. Using functional second-derivative analysis, we further examine how inhomogeneous K 0 affects stability. The transition between frustrated constriction and abscissional elongation is numerically analyzed through conformal solutions to the governing inhomogeneous K 0 − field.