Normal Form for Edge Metrics

TL;DR

This paper develops a normal form for edge metrics by solving a singular eikonal equation, establishing a correspondence with boundary conformal structures, and proving existence and uniqueness of solutions.

Contribution

It introduces a novel normal form for edge metrics using a characteristic Hamiltonian flow approach, linking interior metrics to boundary conformal data.

Findings

Normal form characterized by a 1-1 correspondence with boundary conformal infinity

Existence and uniqueness of smooth solutions to the singular eikonal equation

Construction of a characteristic Hamiltonian flow in the jet bundle

Abstract

A normal form for edge metrics is derived under the necessary conditions that the metric be normalized and exact. The normal forms for such an edge metric are shown to be in 1-1 correspondence with representative metrics for a reduced conformal infinity on the boundary. The normal form is constructed via solution of a singular eikonal equation at infinity. The eikonal equation is solved by proving existence and uniqueness of smooth solutions of a class of characteristic nonlinear first-order initial value problems. This is carried out by constructing a characteristic version of a Hamiltonian flow-out in the first jet bundle of the solution.