# On the conformal method for the Einstein constraint equations

## TL;DR

This paper applies global analysis and degree theory to study solutions of Einstein constraint equations via the conformal method, providing new proofs and insights into existence, multiplicity, and non-existence results.

## Contribution

It introduces a novel application of Smale's degree-theoretic methods to the conformal approach for Einstein constraints, offering new proofs and connections to existing results.

## Key findings

- New proof of Maxwell and Holst-Nagy-Tsogtgerel existence results
- Relation of the conformal method to the limit equation of Dahl-Gicquaud-Humbert
- Non-existence results of Nguyen clarified through this approach

## Abstract

In this work, we use the global analysis and degree-theoretic methods introduced by Smale to study the existence and multiplicity of solutions of the vacuum Einstein constraint equations given by the conformal method of Lichnerowicz-Choquet-Bruhat-York. In particular this approach gives a new proof of the existence result of Maxwell and Holst-Nagy-Tsogtgerel. We also relate the method to the limit equation of Dahl-Gicquaud-Humbert and the non-existence result of Nguyen.