Let $M^2$ be a closed surface, say the 2-sphere. Is there any example of metric on it such that there are uncountably many points are conic and the metric is smooth elsewhere?


We call $p\in M$ a conic point if there exists $\lambda\_i\to \infty$ such that
$(\lambda\_i M, p)$ converge to a linear cone $C(S(\ell))$, the cone over a circle of length $\ell$, for $\ell\ne 2\pi$.


For convex surface, there at most countably many conic points, which is proven by AD Alexandrov.



Closed surface with uncountably many conic points?
The answer is NO.


Fix $\varepsilon>0$ and $\delta>0$.


Consider the set $X\_{\varepsilon,\delta}$ of all the points in $M$ 
such that for any $r<\varepsilon$, the $r$-neighborhood of any $x\in X\_{\varepsilon,\delta}$ is $r{\cdot}\delta$-close 
to $r$-ball in the cone over the circle with length $\ell$ such that $|\ell-2{\cdot}\pi|>\varepsilon$.


Note that $X\_{\varepsilon,\delta}$ is uncountable for small $\varepsilon >0$
and any $\delta>0$.
On the other hand, if $\delta$ is small then the set $X\_{\varepsilon,\delta}$ is discrete, a contradiction.


