Surfaces with radially symmetric prescribed Gauss curvature

We study conformally flat surfaces with prescribed Gaussian curvature, described by solutions $u$ of the PDE: $\Delta u(x)+K(x)\exp(2u(x))=0$, with $K(x)$ the Gauss curvature function at $x\in\RR^2$. We assume that the integral curvature is finite. For radially symmetric $K$ we introduce the notion of a least integrally curved surface, and also the notion of when such a surface is critical. With respect to these notions we analyze the radial symmetry of $u$ for the whole spectrum of possible integral curvature values. Under a mild integrability condition which rules out harmonic non-radial behavior near infinity, we prove that $u$ is radially symmetric and decreasing in the following categories: (1) $K$ is decreasing, $u$ a classical solution, and the integral curvature of the surface is above critical; (2) $K$ is decreasing, $u$ a classical solution, the integral curvature of the surface is critical, and the surface satisfies an additional integrability condition which is mildly stronger than finite integral curvature; (3) $K$ is non-positive. In categories 1 and 2, $K$ is allowed to diverge logarithmically or as power law to $-\infty$ at spatial infinity. Examples of nonradial solutions which violate one or more of our conditions are discussed as well. In particular, for non-positive and non-negative $K$ that satisfy appropriate integrability conditions and otherwise are fairly arbitrary, we introduce probabilistic methods to construct surfaces with finite integral curvature and entire harmonic asymptotics at infinity. For radial symmetric $K$ these surfaces are examples of broken symmetry.