Lemma 4.1.1 of "The heat equation shrinking convex plane curves"

Question

In the article "The heat equation shrinking convex plane curves" by M. Gage and R. S. Hamilton, I didn't understand the reciprocal of the lemma. What is proper curvature and the Gauss map of a strictly convex plane curve? I searched, but I didn't find these definitions. Is it a Gauss map of a strictly convex plane curve just an unit normal vector?

Thanks in advance!

Answer 1

The authors write "has the proper curvature" to mean "has curvature given by the function $k$" - the word proper is being used in the grammatical sense of "appropriate" or "correct", not as a mathematical term.

The Gauss map of a curve $\gamma : S^1 \to \mathbb R^2$ is just a unit normal vector field $N_\gamma : S^1 \to S^1$, yes.