(a) Show that for a compact regular surface $S \subset \mathbb{R}^{3}$, there exists a point $p \in S$ such that $K(p)>0$, where $K$ denotes the Gaussian curvature. Show that if $S$ is contained in a closed ball of radius $R$ in $\mathbb{R}^{3}$, then there is a point $p$ such that $K(p) \geqslant R^{-2}$.

(b) For a regular surface $S \subset \mathbb{R}^{3}$, give the definition of a geodesic polar coordinate system at a point $p \in S$. Show that in such a coordinate system, $\lim _{r \rightarrow 0} G(r, \theta)=0$, $\lim _{r \rightarrow 0}(\sqrt{G})_{r}(r, \theta)=1, E(r, \theta)=1$ and $F(r, \theta)=0$. [You may use without proof standard properties of the exponential map provided you state them clearly.]

(c) Let $S \subset \mathbb{R}^{3}$ be a regular surface. Show that if $K \leqslant 0$, then any geodesic polar coordinate ball $B\left(p, \epsilon_{0}\right) \subset S$ of radius $\epsilon_{0}$ around $p$ has area satisfying

$$\text { Area } B\left(p, \epsilon_{0}\right) \geqslant \pi \epsilon_{0}^{2}$$

[You may use without proof the identity $(\sqrt{G})_{r r}(r, \theta)=-\sqrt{G} K$.]

(d) Let $S \subset \mathbb{R}^{3}$ be a regular surface, and now suppose $-\infty<K \leqslant C$ for some constant $0<C<\infty$. Given any constant $0<\gamma<1$, show that there exists $\epsilon_{0}>0$, depending only on $C$ and $\gamma$, so that if $B(p, \epsilon) \subset S$ is any geodesic polar coordinate ball of radius $\epsilon \leqslant \epsilon_{0}$, then

$$\text { Area } B(p, \epsilon) \geqslant \gamma \pi \epsilon^{2}$$

[Hint: For any fixed $\theta_{0}$, consider the function $f(r):=\sqrt{G}\left(r, \theta_{0}\right)-\alpha \sin (\sqrt{C} r)$, for all $0<$ $\alpha<\frac{1}{\sqrt{C}}$. Derive the relation $f^{\prime \prime} \geqslant-C f$ and show $f(r)>0$ for an appropriate range of $r .$ The following variant of Wirtinger's inequality may be useful and can be assumed without proof: if $g$ is a $C^{1}$ function on $[0, L]$ vanishing at 0 , then $\int_{0}^{L}|g(x)|^{2} d x \leqslant \frac{L}{2 \pi} \int_{0}^{L}\left|g^{\prime}(x)\right|^{2} d x$.]