Is it permissible for a man to only have beard on his chin? And can he trim the beard down to just 2-3mm providing that it is visible that this man has a beard?
It is not permissible for a man to only have beard in his chin as it implies "shaving" which is considered haram. As far as the trimming, the Muslim person should be considered as "having beard" according to common view.
With prayers for your success.