If you're asking how to choose a different center for the plotted map while still staying in Stonyhurst heliographic coordinates, you can change the definition of the reference coordinate. In the gallery example, `frame_out` is the reference coordinate, and the first argument for the `SkyCoord` constructor is the Stonyhurst longitude of the reference coordinate, and will be the center of the map in the horizontal direction.
If you're asking how to switch to Carrington heliographic coordinates in addition, you again would modify the definition of the reference coordinate. For the `SkyCoord` call, you would specify `frame="heliographic_carrington"`. Because of how SunPy implements Carrington heliographic coordinates, you will also need to specify the observer location. To do that, add `observer=aia_map.observer_coordinate` to the `SkyCoord` call.
Carrington longitude: `sunpy.coordinates.sun.L0()`
Carrington rotation number: `sunpy.coordinates.sun.carrington_rotation_number()`
Both of the above functions are for Earth center. I'll also note that SunPy's definition Carrington longitude is slightly different from some other sources:
> A related question, ...
My guess is that your map's definition of the physical radius of the Sun does not quite match the IAU 2015 value (695.7 Mm). This is not surprising: for example, AIA and HMI maps use a rounded value (696 Mm). In fact, the gallery example makes the same error and needs to be fixed. You'll note that the gallery example has a similar "tapered" look, and it shouldn't. I've occasionally seen this type of error manage to preserve a "fringe" like what you show. (The gap between the fringe and the main area should be reprojectable.)
If you're running the latest release of SunPy, you can add the keyword `rsun` to define the non-standard radius:
`frame_out = SkyCoord(..., rsun=aia_map.coordinate_frame.rsun)`
If you're running an earlier version of SunPy, the most expedient solution is to overwrite your map's metadata to use the IAU 2015 value.
Setting `rsun` should fix most, if not all, of the issue. There's a chance that pixels right at the limb may not be able to reproject as desired. In those cases, you may be able to improve the results by using `sunpy.coordinates.Helioprojective.assume_spherical_screen(..., only_off_disk=True)`.
Let me know if you need further help with any of the above.
Thanks,
Albert