The Liouville integral is a parametric representation of s surface with constant Gaussian curvature. We start with fundamental properties of surfaces, and then derive this Liouville integral via complex variables. The representation formula, surprisingly, induces a quantized blowup mechanism of the Boltzmann–Poisson (BP) equation because of its geometric feature of nonlinearity, associated with a flat plane and a sphere. The complex analysis used, furthermore, leads to a new insight into the statistical mechanics, that is, recursive hierarchy in the context of Onsager’s theory on many point vortices, as the location of blowup points of the family of solutions to the BP equation is controlled by the Hamiltonian concerning the system of point vortices. We thus reach a fusion of the theories of surface, complex variables, elliptic equations, and statistical mechanics through BP, or Gel’fand equations in two-space dimensions. These phenomena encountered in the interaction of analysis, geometry, and physics, spread into a wide area in the theory of nonlinear partial differential equations, as is described in later chapters. In this chapter, after a summary in the first section, we confirm the classical surface theory in the second section. The Liouville integral is induced in the third section, which leads to a motivation of the study of BP equation in the final section, that is, the quantized blowup mechanism and recursive hierarchy.