Let $S \subset \mathbb{R}^{3}$ be a surface.

(a) Define the Gauss Map, principal curvatures $k_{i}$, Gaussian curvature $K$ and mean curvature $H$. State the Theorema Egregium.

(b) Define what is meant for $S$ to be minimal. Prove that if $S$ is minimal, then $K \leqslant 0$. Give an example of a minimal surface whose Gaussian curvature is not identically 0 , justifying your answer.

(c) Does there exist a compact minimal surface $S \subset \mathbb{R}^{3}$ ? Justify your answer.