The area where a gutter would be is called an eavesdrop on buildings that don't have gutters. It is the area underneath the eaves (overhang) of the roof where water drips off. It is also called an eavesdrip sometimes. A lot of people have never heard the word eavesdrop used in this way. They just assume it is the act of eavesdropping (listening in on others conversation) thus the confusion with yo
Whoops! Sorry, I dropped an a! Glad you caught it. There are perhaps 50 company names that use the Evestrough spelling, however. That's Canada for you!
Well! Then this may be the closest thing to full answer to the original question:
A gutter (more specifically a rain gutter) is a trough mounted at the eaves of a roof, to catch and redirect rainwater so that it doesn't pour directly onto the ground below.
In Canadian English, the common word for a gutter is an eavestrough.