WHAT HAPPENS IF BLACK DOES NOT PROMOTE?

White threatens Rb1+ Kf2 and Rb2 captures the pawn.
Black can prevent this by playing Bd4, but with a rook on the first rank black will never be able to promote.

I predict Nb6 mate with the black knight on b5.

Nc1 forces Black to do something or else lose his Knight with Kxb1 and quickly fall apart. Na3 is the only response to this, and then this continuation occurs:

1. Nc1 Na3
2. Na2+ Ka4
3. Nc3+ Kb4
4. Nd5+ Ka4
5. Nb6+ Kb4
6. Bc3+ Kb5
7. Nd5 Ka4
8. Be1

Now Black is in Zugzwang.

King can't move to it's only square b5 but it would just lead to Kxa3 obviously.

Knight can't move to b1 or c2 without being taken by the King, and Nb5 isn't possible due to Nb6++.

c3+ and e3 are responded to with Nxc3+ and Nc3+, respectively. After that, if Ka5, White just responds with Kxa3. If Kb4 instead, White responds with Nb1+ (discovered check), followed by Nxa3.

So with Black only having 1 or 2 pawns, it's now all just a matter of rounding them up with a bit of technique, and then it's just a King + Bishop + Knight vs. King Endgame, which is easily doable.

Really fun Endgame study this week. Hope for another fun one next week.

Oh wait, my bad, 7. should be Nd7.

I think you can play 5. Bc3 already

Note: The new (proper) continuation is now:

1. Nc1 Na3
2. Na2+ Ka4
3. Nc3+ Kb4
4. Nd5+ Ka4
5. Nb6+ Kb4
6. Bc3+ Kb5
7. Nd7 Ka4 (any other move just leads to the Knight being taken)
8. Nc5+ Kb5
9. Nxe4 Ka4 (again, any other move leads to the Knight being taken).

Now there are only Pawn is taken so the game is lost for Black. Continuation if you want to continue:

10. Nc5+ Kb5
11. Nd7 Ka4 (if Kc6, then Ne5 followed by Kxa3).
12. Nb6 Kb5
13. Bd4 Kb4
14. Nd5+

and now the Black Knight is lost. Ka5 and Kb5 are responded to with Kxa3, and Ka4 (maintaining protection of the Knight) is just responded to with:

14. Nd5+ Ka4
15. Bc3

And NOW Black is in Zugzwang. He can't advance the Pawn, so he only has 1 King and 3 Knight moves available. King moves anywhere, Kxa3 occurs. Knight moves to b1 or c2 and it's taken with Kxb1 or Kxc2. And Nb5 isn't possible due to Nb6++. So all the Black king has is a Pawn now, which is being blockaded by the Bishop and the White King can easily force the Black King away from it take it with the King and Knight while protecting the Bishop with his King. NOW it just becomes a King + Bishop + Knight vs. King Endgame.

Cak, this continuation is necessary to ensure the Black Knight remains trapped on a3 and e pawn stays and is captured on e4 and that Black does not have any strategic breaks in which it can advance. If 5. Bc3, then just e3 and the e pawn can't be taken due to the Knight finally being able to escape with Nb5, leading to a draw.

You might be right, but I was thinking 5. Bc3 e3 6. Be1 e2 (6...c3+ 7. Nxc3+ Kb4 8. Nb1+) 7.Bc3 and black will run out of moves. Even 7.Nc3+ seems possible.

Your continuation would be followed with:
7. Bc3 e1 Queening.
8. Bxe1 c3+

Now even with Nxc3+, the Black Knight will still be able to escape and force White to exchange the Bishop or the Knight, leading to a draw. Remember, this game is to play and WIN, and Black can get away with simply trying to exchange the Knight for the White Bishop or Knight. Your Nc3+ continuation would lead to:

7. Nc3+ Kb4

Now what? Moving the Knight back with Nd5+ (double check) and the King would simply move back with Ka4, repeating the position. The White Knight can't move to any other square even with the discovered check due to any Knight move will be refuted with c3+ (since it's no longer blockaded), forcing a capture or for the White King to move and allowing the Black Knight to escape with Nb5. And if the White Knight doesn't move, again, the Black Knight would be able to escape with Nb5 anyway, eventually forcing an exchange.

The goal of this endgame is to maintain the entrapment of the Black Knight, tying the Black King down to it. While you're doing that, get the E pawn before it has a chance to advance, maintain a blockade of the C pawn, and also maintain the constant threat of Nb6++. 5. Bc3 eventually fails to do all of them and allows the Black Knight to escape.

In all the variations you mention, white can give a discovered check Nb1+ and win the black knight.

Again, like I specifically said before, Nb1+ would simply be refuted with c3+ (it blocks the discovered check), forcing a capture or the King to move. The only move that keeps the pressure is Bxe3+. And if you do Bxe3+?

7. Nc3+ Kb4
8. Nb1+ e3+
9. Bxe3+ Ka4

Think the Knight can be taken now? Think again, the E pawn is no longer blockaded.

10. Nxa3 e1 Queening.
11. Bxe1
STALEMATE. The Black King no longer has any spaces to move.

The only other space the White Knight can move is Nd2, which it can't do due to blocking the Bishop from taking the Promoted Pawn. And if the White Knight doesn't move, the Black Knight will simply take it and force a draw. As I've proven, e3+ will be a proper response to any Knight move, even with the discovered check. Except Nd5+ (Double Check), but then Kb4 and now you're just repeating the position.

And of course, capturing after e3+ with the Knight would simply allow the Black Knight to escape, eventually forcing an exchange and a draw.

The stalemate motive its just brilliant, and gives the study much more depth.