Can a positive curvature smooth metric on the disk be $C^0$ approximated by smooth flat metrics?


Similar question under extra assumption: the same conformal class.


If there are positive examples in both cases, then are there some relatively weak extra-assumptions that the answer is no? Also I am curious how the answer depends on the disk dimension.



Continuous approximation of smooth metrics.
No, because too many geometric quantities are continuous in the $C^0$ topology: e.g. length of curves, volume.


Take the standard metric on the radius $1$ sphere, remove a small open disk near the south pole, and push forward stereographically the metric to a metric on the disk. Then the length of the boundary in this metric is some $\ell\ll 1$, and the volume of the metric is close to $4\pi$. Any sufficiently $C^0$-close metric $\tilde g$ must have volume close to $4\pi$ and give to the boundary length much smaller than $1$. This prevents it to be flat, as it would violate the Euclidean isoperimetric inequality.


