Equivalence between the Curve Shortening Flow and a parabolic PDE with a Boundary Value Problem

Question

I'm reading The heat equation shrinking convex plane curves and I'm trying understand the theorem $4.1.4$ that asserts that the Curve Shortening Flow is equivalent to a parabolic PDE with a Boundary Value Problem. I didn't understand why the items $(i)$ and $(iii)-(a)$ of the theorem is valid considering the Curve Shortening Flow. I think it's a consequence direct of a result in Parabolic PDEs Theory, but I'm not very familiar with this area. Could anyone explain with details how can I prove $(i)$ and $(iii)-(a)$?

Thanks in advance!