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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 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?

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How are you integrating a function that takes values in $\mathbb T$? $\mathbb T$ isn't a vector space, and what measure on $\mathcal T$ are you integrating with respect to? – Ryan Budney Feb 25 '13 at 8:25
It sounds like you're asking the question, which functions $S^1 \times S^1 \to \mathbb R$ can be realized as Gauss curvatures for some Levi-Cevita connection on $S^1 \times S^1$ ? – Ryan Budney Feb 25 '13 at 8:28
@Ryan: Sorry, this was a typo, it should have been $\mathbb{R}$. – Hans Stricker Feb 25 '13 at 8:42
@Ryan: Is "Levi-Civita connection on $S^1\times S^1$" the same as "homeomorphic to $S^1\times S^1$"? If so: yes, that's what I am asking. – Hans Stricker Feb 25 '13 at 8:44
If you don't demand some regularity of your manifold your curvature could be a distribution rather than a continuous function. That's why I'm asking what you mean by curvature -- curvature of what? If it's the curvature of a Levi-Cevita connection presumably we're talking about smooth manifolds and fairly standard notions of curvature. Your insistence on talking about topological manifolds is what's throwing me now. – Ryan Budney Feb 25 '13 at 8:54
up vote 7 down vote accepted

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.

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