A geometric reduction method for some fully nonlinear first-order PDEs on semi-Riemannian manifolds

TL;DR

This paper introduces a geometric reduction technique using transnormal functions on semi-Riemannian manifolds to transform certain nonlinear first-order PDEs into ODEs, enabling the construction of solutions with specific geometric properties.

Contribution

The paper presents a novel reduction method for fully nonlinear first-order PDEs on semi-Riemannian manifolds using transnormal functions, leading to new solution constructions.

Findings

Reduced PDEs to ODEs using transnormal functions

Established local existence of solutions constant along level sets

Derived new solutions to geometric eikonal equations

Abstract

Given a semi-Riemannian manifold $(M,\langle \cdot,\cdot\rangle_g),$ we use the transnormal functions defined on $M$ to reduce fully nonlinear first order PDEs of the form \[ F(x,u,\langle \nabla_g u, \nabla_g u \rangle_g) = 0,\qquad \text{on }M \] into ODEs and obtain local existence results of solutions which are constant along the level sets of the transnormal functions. In particular, we apply this reduction method to obtain new solutions to eikonal equations with a prescribed geometry.