I'm doing well, Mike, thank you!
You're right in thinking that something's grammatically missing in this sentence: "
If interested, kindly send me a message." Indeed, as you suggest, the scrupulously correct way to construct that sentence is this: "
If you are interested, kindly send me a message." However, in their best conversational English, most native English speakers--confidently and without batting an eyelash about its syntax--actually would say or write that sentence without the subject "you" and the verb "are" in the "if clause: "If interested, kindly send me a message."
What you've stumbled upon is what's known in English as the
elliptical sentence. It is a form of a sentence that knocks off some of its words or phrases for brevity’s sake, taking for granted that the reader or listener—aware of the context—would just logically fill in the gaps with the missing grammatical elements. For instance, before being ellipted, a sentence may read this way: “You
may go when you’re done
with your weekly report.” That sentence can drop the words “you may” and “with your weekly report assignments” to come up with this ellipted, short-and-sweet statement: “Go when you’re done.”
To form an elliptical sentence, typically dropped are the subject and verb in the conditional clause, as what happened to both "you" and "are" in the sentence you presented: "If interested, kindly send me a message." You incorrectly surmised that the form of the verb in "if"-clause of that ellipted sentence is "interested." In fact, that word isn't a verb but an adjective, which is all that remains of the "if"-clause after both its subject "you" and verb "are" were ellipted or dropped. Typically, a clause is ellipted when both its subject and operative verb are dropped without changing or substantially altering its intended sense or meaning.
For a better understanding and appreciation of elliptical sentences, I am providing links to several discussions about it in the Forum over the years.
FURTHER READINGS ON ELLIPTICAL SENTENCES:Elliptical sentences often read and sound better than regular sentences (2010)Deconstructing and understanding those puzzling elliptical sentences (2011)The difference between elliptical sentences and elliptical clauses (2018)
