I was talking to a professor about Morse theory and he said that there a result similar to:
Two bodies of water in a domain that are currently not connected will connect across a saddle point.
I cannot remember if this was a result of the mountain pass lemma or of some theory in Morse theory (We were chatting about many different things). This is not my area of math and I was hoping someone could point me to a theory statement like this somewhere so I can read more about it?
Thank you.
If you know that the domain is non-degenerate you can prove this through Morse theory. If you don't know that the critical points are non-degenerate it is a bit harder to say what a saddle point is: but the mountain pass lemma will provide a critical point which has saddle like behavior. Anyway to prove this in the non-degenerate situation you should play around with the morse relations:
Let $c_i$ be the critical poins of index $i$ and $b_i$ the betti numbers. If your domain is connected $b_0=1$ and some sublevel set is not then $c_0\geq 2$. Then the second morse relation states
$$ c_1-c_0\geq b_1-b_0 $$ So $c_1\geq b_1+c_0-b_0\geq 1.$
Hence there must be a critical point of index $1$, i.e. there is a saddle.