The heat equation shrinking convex plane curves 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.
 A: The key of this theorem is to show that the curves under the flow do not generate singularities until the area becomes zero.
This is done by summarizing all the previous results of that section:


*

*Provided that the area is bounded away from zero, they obtain bounds
for $\kappa$ and its spatial derivatives (this is done in sections
4.3 and 4.4 of the paper). Using the evolution equation ${\partial\kappa\over\partial
   t}=\kappa^2{\partial^2\kappa\over\partial\theta^2}+\kappa^3$ they
bound the temporal derivatives as well.

*Now, if you suppose that there exists a solution in $[0,T)$ to the
equivalent problem to the CSF proposed in 4.1.4 theorem and that
the area doesn't get to $0$ ($\lim_{t\to T}A(t)>0$), then there
exists the limit for $\kappa$, when $t$ goes to $T$, which is
$\mathcal{C}^\infty$ and we can extend the solution past $T$.

*Finally, the solution $\kappa$ to the equation
${\partial\kappa\over\partial
   t}=\kappa^2{\partial^2\kappa\over\partial\theta^2}+\kappa^3$ can be
transformed into a solution of the heat equation for curves using
4.1.1 lemma.


Hope this clarifies the procedure a little bit.
