Bounds on area of geodesic disk Are there upper and lower bounds, in terms of curvature, on the area of geodesic disks on compact smooth 2-manifolds embedded (or immersed) in $\mathbb{R}^3$?
It is tempting to imagine bounds must exist in terms of minimum and maximum Gauss curvature, and perhaps this is true for sufficiently small disks (that fit inside a single chart), but a simple counter example is a very thin pill-shaped tube. Still, this example leaves open the hope for bounds in terms of e.g. both principal curvatures.
 A: Upper bounds are provided by the Bishop-Gromov volume comparison theorem: If the curvature of $M$ is bounded below by $K_{\rm min}$, then the volume of a geodesic ball $B(p,r)\subset M$ is bounded above by the volume of the a corresponding ball $B(q,r)\subset N$ where $N$ is the space of constant curvature $K_{\rm min}$.
Such a global comparison (using Gaussian curvature) for lower bounds is impossible for the reason you mention, but you can still get local control by looking at small enough balls. Working in normal coordinates you can arrive at a series expansion for the geodesic ball volume that looks (in your 2-dimensional case) something like
$$ V(B(r)) = \pi r^2 - C K r^4 + O(r^6)$$
where is $C$ is some constant universal to all $2$-manifolds - see e.g. this Wikipedia article if you want to work out what it is. Thus for every $\epsilon \in (0,1)$ there is a $\delta > 0$ such that
$$ \pi r^2 - \epsilon C K_{\rm max} r^4 \le V(B(r)) \le \pi r^2 - \epsilon C K_{\rm min} r^4 $$
for $r \in (0,\delta)$.
