On Two Dimensional Flat Hessian Potentials
Hanwen Liu
TL;DR
The paper addresses the construction and characterization of two-dimensional flat Hessian potentials, i.e., potentials whose Hessians yield flat metrics on surfaces. It develops a PDE-based framework by encoding Hessian data into a hydrodynamic system and applying a hodograph transform, which reduces the problem to a Klein-Gordon equation with a time-dependent potential and is solvable by separation into eigenmodes. The authors obtain a large family of explicit flat Hessian potentials, including a canonical Euclidean-plane example and homogeneous-degree cases, with a procedure to recover the potential by integrating the Hessian. By connecting Hessian geometry with integrable-systems techniques, the work provides explicit constructions of flat Hessian representations in two dimensions and broadens the toolkit for differential-geometric analysis and applications in mathematical physics.
Abstract
A Riemannian metric is termed a Hessian metric if in some coordinate system it can be locally represented as the Hessian quadratic form of some locally defined smooth potential function. Under very mild extra technical conditions, we first theoretically describe the potentials of flat Hessian metrics on surfaces, and then construct these potentials explicitly using methods from integrable systems.