lot depends on your water temperature. As long as your water temperature stays
at or above 45 degrees, you'll be getting some benefit from your filtration and
your filters and pumps should be allowed to run. Your fish will be active enough
to need the filtration. If your water temps drop below 45 degrees, all you are
getting is mechanical filtration and since you won't be feeding the fish, biological
filtration isn't as much of an issue.
circulation over your falls will maintain some oxygenation, especially if you've
got a lot of turbulence and water movement.
The point at
which the falls become a liability is the point when your air temperature drops
below freezing and stays there. Water over the falls will tend to freeze in interesting
patterns, with the real possibility that your water will be diverted out of the
pond, leaving your fish stranded.
I suspect that long-term
freeze is rare in North Carolina. Run your pumps as long as you can.