Gaussian curvatures of Lorentzian metrics on the plane and punctured planes

Abstract:

We prove that every $f \in {C^k}\left ( {{R^2}} \right )$ is the Gaussian curvature of some ${C^{k + 1}}$ -Lorentzian metric $\left ( {0 \leq k \leq \infty } \right )$. Let $M$ denote the cylinder. We prove that every continuous function on $M$ is the Gaussian curvature of some ${C^1}$-Lorentzian metric. If $f \in {C^k}\left ( M \right )$ satisfies the condition (H) in the Lemma 2 below, then it is the curvature function of some ${C^{k + 1}}$-Lorentzian metric. If $f \in {C^k}\left ( {{R^2}} \right )\left ( {1 \leq k \leq \infty } \right )$ has compact support, then the Lorentzian metric can be made complete.