My real question is that I notice when I mow the smaller gullies still
are free of grass even though they are not very deep (1/2 inch at most
and an 1-2 inches in width). The grass around them is thick and green.
Given the grass is growing now why do these areas not fill in?

I read a lawn care book from Ortho, and it says that we need to mow
the lawn often when the grass is actively growing in order to
encourage it to grow out instead of growing tall. If yours is cool
season grass, the "active-growing" season is spring and fall. If this
is the case, you may need to mow the grass often to keep the grass low
enough (even twice a week). If my memory is right, you are better off
doing this (mowing often) in spring (rather than fall). Unfortunately,
I don't have the book with me right now, and I cannot verify if fall
is OK to mow often or not.

I don't remember how low you should mow the grass. I only remember
that you need to mow often. You may need to check this out in the book
store.

I haven't tried this yet. I intend to do just that in next spring (or
may be this fall).

Hope this helps.

Jay Chan