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It is a well known fact that locally CAT(-1) metrics on surfaces can be approximated by hyperbolic polyhedral metrics with cone singularities: roughly speaking you pick a geodesic triangulation of the surface and replace each triangle with an hyperbolic one.

Can we do better? More precisely, given a locally CAT(-1) distance $d$ on a closed surface, can we find a sequence of smooth metrics with curvature less than -1 such that the induced distance corverges to $d$? Or equivalently, can a polyhedral hyperbolic metric with cone angles bigger than $2π$ be approximated by a sequence of smooth metrics?

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    $\begingroup$ You don't have any CAT$(-1)$ distance on any closed surface. Do you mean locally CAT$(-1)$? $\endgroup$
    – YCor
    Mar 26, 2018 at 8:16

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This is getting to long for a comment :

First you only need to work locally around each vertex of your polyhedron which where the metric is a hyperbolic cone metric.

Locally, an hyperbolic cone metric can be written in normal coordinates as $dr^2+(a\sinh(r))^2d\theta^2$ with $a\geq 1$ (the cone angle is then $2\pi a$).

To get your approximation, all you need to do is to approximate $f(r)=a\sinh(r)$ in $C^0$ by functions $f_i$ such that :

  • $f_i=f$ outside a neighborhood of $0$.

  • $f_i(r)\sim r$ as $r\to 0$ (to get a smooth metric).

  • $-\frac{f_i''}{f_i}\leq -1$ (this is the condition on the curvature).

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