The authors’ real concern is not the future boundaries within the UK or the following economic recession , but the stability of the whole European Union if one if its “big tree” leaves.
or
The authors’ real concern are not the future boundaries within the UK or the following economic recession , but the stability of the whole European Union if one if its “big three” leaves.
Verb coordination seems really difficult in such examples! Is there any general rule so I don't have to bother you again and again?
The authors’ real concern is/are not the future boundaries within the UK or the following economic recession , but the stability of the whole European Union if one if its “big three” leaves.
In almost all cases, the form of the verb (singular or plural) is dependent on whether the subject is singular or plural. So the first thing to do is identify the subject, the phrase that is controlling the verb.
In your example, the subject must be the noun phrase "the authors' real concern", since that is the phrase immediately before the verb. Now you have to decide which word in that phrase is the head word, the most important word, and the one that ultimately determines the verb-form. Is it plural "authors" or singular "concern"? To help you decide, compare these two sentences:
"Fred's car is very nice."
"Fred's cars are very nice".
Notice that in both cases, it's the 'number' of the car that is determining the verb-form. It can't be Fred's because that has remained unchanged. So now we know that "car(s) is the head word in the phrase; the one that determines the verb-form
Now apply that procedure to your example and tell me what you think is the head word in the subject phrase in your example. Is it plural "authors" or singular "concern"? Tell me what you decide.
Incidentally, there is another way to find the head word in examples like yours. Let me show you. Here's your original sentence again; it's a statement:
The authors’ real concern is/are not the future boundaries within the UK or the following economic recession , but the stability of the whole European Union if one if its "big three" leaves.
Now turn the first part into a question:
(1) Are the authors the future boundaries within the UK....?
(2) Is the real concern the future boundaries within the UK...?
Clearly, (1) is ungrammatical nonsense, but (2) is fine, thus again proving that the singular noun "concern" is the head word, so singular "is" must be correct.
Actually I forgot to mention 2 things, the correct version is
The author's real concern...
and, second, I was thinking about the phrase "the future boundaries within the UK...", whether there should be coordination between the Plural "boundaries" and the verb, this was my initial hesitation. Sorry for being too slow, figuratively.