Application of isotropic geometry to the solution of the Monge–Ampere equation

Abstract

This paper explores the Monge–Ampere equation in the context of isotropic geometry. The study begins with an overview of the fundamental properties of isotropic space, including its scalar product, distance formula, and the nature of surfaces and curvatures within this geometric framework. A special focus is placed on dual transformations with respect to the isotropic sphere, and the self-inverse property of the dual surface is established. The article formulates the Monge–Ampere equation for isotropic space and studies its invariant solutions under isotropic motions. Several lemmas are proved to demonstrate how solutions transform under linear modifications and isotropic motions. A specific class of Monge–Ampere-type nonlinear partial differential equations is solved analytically using dual transformations and separation of variables. Additionally, translation surfaces and their curvature properties are studied in detail, particularly through the lens of dual curvature. The results demonstrate the deep relationship between curvature invariants and Monge–Ampere-type equations and show how duality simplifies the solution of nonlinear PDEs. These methods can be used for surface reconstruction and modeling in isotropic spaces.