Recent models of straight diffusion flame edges are extended to consider the effect of a curved edge forming the perimeter of an axisymmetric ‘hole’, where a burning flame surrounds a quenched inner region. For ‘free’ flame-holes (without a heat sink near the axis), at small Damköhler number (Da), the holes grow if the initial radius is large but shrink if it is small. For largeDa, the holes shrink for any initial radius. Thus, free flame-holes are not stable for anyDa, which is consistent with experimental observations. When the flame-hole is ‘anchored’ by a heat sink near to the axis, stationary holes of finite radius can exist for sufficiently highDa, but the solutions revert to ‘free’ hole behaviour for radii sufficiently larger than the heat sink radius. Based on these results, it is suggested that quasi-stationary flame-holes are not likely to be a common feature of turbulent diffusion flames, except possibly when large lateral gradients ofDaare present due to intense vortices passing through the flame front.