Very large-mode-area ${\mathrm{Yb}}^{3+}$ -doped single-mode photonic bandgap (PBG) Bragg fiber oscillators are considered. The transverse hole-burning effect is numerically modeled, which helps properly design the PBG cladding and the ${\mathrm{Yb}}^{3+}$ -doped region for the high-order mode content to be carefully controlled. A ratio of the ${\mathrm{Yb}}^{3+}$ -doped region diameter to the overall core diameter of 40% allows for single-mode emission, even for small spool diameters of $15\mathrm{cm}$ . Such a fiber was manufactured and subsequently used as the core element of a cw oscillator. Very good beam quality parameter $M^2=1.12$ and slope efficiency of 80% were measured. Insensitivity to bending, exemplified by the absence of temporal drift of the beam, was demonstrated for curvature diameter as small as $15\mathrm{cm}$ .