We report on a newly designed and fabricated ytterbium-doped large mode area fiber with an extremely low NA (~0.04) and related systematic investigations on fiber parameters that crucially influence the mode instability threshold. The fiber is used to demonstrate a narrow linewidth, continuous wave, single mode fiber laser amplifier emitting a maximum output power of 3 kW at a wavelength of 1070 nm without reaching the mode-instability threshold. A high slope efficiency of 90 %, excellent beam quality, high temporal stability, and an ASE suppression of 70 dB could be reached with a signal linewidth of only 170 pm.