The Subjunctive Mood is used in adverbial clauses of concession. Adverbial clauses of concession are introduced by the conjunctions and connectives though, although, however, no matter, whatever, whoever, etc. The analytical subjunctive with the mood auxiliary may (might) is generally used.

Though he may (might) be tired he will go to the concert.

Tired though he may (might) be he will go to the concert.

No matter how tired he may (might) be he will go to the concert.

However tired he may (might) be he will go to the concert.

Whatever obstacles may arise, we shall not give in.

You’re my daughter, Alice. Although you may never admit it, you’re like me. (Gow and D’Usseau)

Remember, the truth, however ashamed of it you may be, is better than any lie. (Dreiser)

I must return to the city, no matter what dangers may lurk there. (Dreiser) 

…I should like to do some good to you and your husband, whoever he may be. (Hardy)

If the action of the subordinate clause is prior to, that, of, the principal clause the Perfect Infinitive is generally used.

However badly he may have behaved to you in the past he is still your brother. (Wilde)

The Indicative Mood is used in adverbial clauses of concession when the action or state is considered as an actual fact and not as something supposed.

Compare:

Cold though it may be, we shall go to the skating rink (Subjunctive Mood).

Cold as it is, we shall go to the skating rink (Indicative Mood).