In the article "The heat equation shrinking convex plane curves" by M. Gage and R. S. Hamilton, I didn't understand the following theorem:
Theorem: the solution continues until the area goes to zero.
Why do we have a limit for $k$ when $t \to T$ ?
As long as the area is bounded away from zero, we get bounds on $k$ and all its derivatives. Using the evolution equation we can bound the time derivatives too. Suppose the solution exists on the interval $[0,T)$ and the area does not go to zero $(\lim \limits _{t \to T} A(t) > 0)$. Then $k$ has a limit as $t \to T$ which is $C^{\infty}$ and we can extend the solution past $T$ and the solution $k$ can be converted to a solution of the heat equation.