A pointwise finite-dimensional reduction method for a fully coupled system of Einstein-Lichnerowicz type

TL;DR

This paper develops a novel finite-dimensional reduction method to analyze non-compactness phenomena in the fully coupled Einstein-Lichnerowicz system, using asymptotic analysis, fixed-point, and ping-pong techniques.

Contribution

It introduces a pointwise finite-dimensional reduction approach for the Einstein-Lichnerowicz system, addressing non-variational challenges with advanced asymptotic and fixed-point methods.

Findings

Constructed non-compactness examples in the focusing case

Developed a fixed-point procedure on blow-up remainders

Combined finite-dimensional reduction with ping-pong method

Abstract

We construct non-compactness examples for the fully coupled Einstein-Lichnerowicz constraint system in the focusing case. The construction is obtained by combining pointwise a priori asymptotic analysis techniques, finite-dimensional reductions and a fixed-point argument. More precisely, we perform a fixed-point procedure on the remainders of the expected blow-up decomposition. The argument consists of an involved finite-dimensional reduction coupled with a ping-pong method. To overcome the non-variational structure of the system, we work with remainders which belong to strong function spaces and not merely to energy spaces. Performing both the ping-pong argument and the finite-dimensional reduction therefore heavily relies on a priori pointwise asymptotic techniques.