Consider the family $\mathbb{T}$ of compact oriented surfaces homeomorphic to the torus $\mathcal{T} = S^1 \times S^1$. Consider arbitrary continuous mappings $k: \mathcal{T} \rightarrow \mathbb{R}$ which obey the condition $\int\_\mathcal{T} k = 0$. 



> 
> For which curvature-like mappings $k$
>  does exist a surface $T \in
> > \mathbb{T}$ which $k$ is the Gaussian curvature of?
> 
> 
> 


For the [topological sphere](https://mathoverflow.net/questions/122835/possible-curvatures-of-the-topological-sphere) the answer seems to be: for all. Can the proof of the latter - which I do not know - be generalized for arbitrary compact oriented surfaces? Or is it already a general proof for arbitrary surfaces?



Possible curvatures of the topological torus
The answer is known for all smooth surfaces:
M. S. Berger, Riemannian structure of prescribed Gaussian curvature for compact 2-manifolds, J. Differential Geom. 5 (1971), 325-332.
J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. 99 (1974), 14-47.
(This deals with the case of torus).


All these results were generalized by Troyanov, 
Prescribing curvature on compact surfaces with conical singularities. 
Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821, who deals with surfaces with singularities,
and in another paper, with open surfaces.


