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The shape equation and linking conditions for a vesicle with two-phasedomains are derived. We refine the conjecture on the general neck condition forthe limit shape of a budding vesicle proposed by J\"{u}licher and Lipowsky[Phys. Rev. Lett. \textbf{70}, 2964 (1993); Phys. Rev. E \textbf{53}, 2670(1996)], and then we use the shape equation and linking conditions to provethat this conjecture holds not only for axisymmetric budding vesicles, but alsofor asymmetric ones. Our study reveals that the mean curvature at any point onthe membrane segments adjacent to the neck satisfies the general neck conditionfor the limit shape of a budding vesicle when the length scale of the membranesegments is much larger than the characteristic size of the neck but still muchsmaller than the characteristic size of the vesicle.

General neck condition for the limit shape of budding vesicles