Non-compactness and infinite number of conformal initial data sets in high dimensions

TL;DR

This paper demonstrates that in high-dimensional closed Riemannian manifolds, the Einstein-Lichnerowicz equation can have infinitely many positive solutions, indicating non-compactness of the solution set.

Contribution

It constructs specific background coefficients on high-dimensional manifolds where the Einstein-Lichnerowicz equation exhibits non-compactness and infinitely many solutions.

Findings

Existence of non-compact solution sets in high dimensions

Construction of background coefficients leading to infinite solutions

Implication for conformal initial data in general relativity

Abstract

On any closed Riemannian manifold of dimension greater than $7$, we construct examples of background physical coefficients for which the Einstein-Lichnerowicz equation possesses a non-compact set of positive solutions. This yields in particular the existence of an infinite number of positive solutions in such cases.