(a) State the Gauss-Bonnet theorem for compact regular surfaces $S \subset \mathbb{R}^{3}$ without boundary. Identify all expressions occurring in any formulae.

(b) Let $S \subset \mathbb{R}^{3}$ be a compact regular surface without boundary and suppose that its Gaussian curvature $K(x) \geqslant 0$ for all $x \in S$. Show that $S$ is diffeomorphic to the sphere.

Let $S_{n}$ be a sequence of compact regular surfaces in $\mathbb{R}^{3}$ and let $K_{n}(x)$ denote the Gaussian curvature of $S_{n}$ at $x \in S_{n}$. Suppose that

$$\limsup _{n \rightarrow \infty} \inf _{x \in S_{n}} K_{n}(x) \geqslant 0$$

(c) Give an example to show that it does not follow that for all sufficiently large $n$ the surface $S_{n}$ is diffeomorphic to the sphere.

(d) Now assume, in addition to $(\star)$, that all of the following conditions hold:

(1) There exists a constant $R<\infty$ such that for all $n, S_{n}$ is contained in a ball of radius $R$ around the origin.

(2) There exists a constant $M<\infty$ such that $\operatorname{Area}\left(S_{n}\right) \leqslant M$ for all $n$.

(3) There exists a constant $\epsilon_{0}>0$ such that for all $n$, all points $p \in S_{n}$ admit a geodesic polar coordinate system centred at $p$ of radius at least $\epsilon_{0}$.

(4) There exists a constant $C<\infty$ such that on all such geodesic polar neighbourhoods, $\left|\partial_{r} K_{n}\right| \leqslant C$ for all $n$, where $r$ denotes a geodesic polar coordinate.

(i) Show that for all sufficiently large $n$, the surface $S_{n}$ is diffeomorphic to the sphere. [Hint: It may be useful to identify a geodesic polar ball $B\left(p_{n}, \epsilon_{0}\right)$ in each $S_{n}$ for which $\int_{B\left(p_{n}, \epsilon_{0}\right)} K_{n} d A$ is bounded below by a positive constant independent of $n$.]

(ii) Explain how your example from (c) fails to satisfy one or more of these extra conditions (1)-(4).

[You may use without proof the standard computations for geodesic polar coordinates: $E=1, F=0, \lim _{r \rightarrow 0} G(r, \theta)=0, \lim _{r \rightarrow 0}(\sqrt{G})_{r}(r, \theta)=1$, and $\left.(\sqrt{G})_{r r}=-K \sqrt{G} .\right]$