I am interested in what possible (local) restrictions there are for Gaussian curvature in dimension 2. More precisely, given a function $f: D^2 \rightarrow \mathbb{R}$, is there a metric on $D^2$ with Gaussian curvature $f$?
An even more local version is the following: given a germ of a function $f$ around the origin in $D^2$, is there a germ of a metric with Gauss curvature f? That is, given $f: D^2 \rightarrow \mathbb{R}$, is there an open subset $U$ around the origin and a metric on $U$ with Gaussian curvature $f|_U$?
The local question is settled by DeTurck in 1980, see his announcement "The equation of prescribed Ricci curvature", Proposition 4.1.
A proof appeared in his paper "Metrics with prescribed Ricci curvature", Seminar on Differential Geometry, pp. 525–537, Ann. of Math. Stud., 102, Princeton Univ. Press, Princeton, N.J., 1982.
The problem is open when you want to construct a metric in the given conformal class. However, if your curvature function is real-analytic, then local existence (in the given conformal class) follows from Cauch-Kovalevskaya theorem.