Improving the recent results for the Vacuum Einstein conformal constraint equation by using the half-continuity method

Dahl-Gicquaud-Humbert showed that if the mean curvature \(\tau\) has constant sign, at least one of the conformal equations and a certain limit equation has a solution. Based on the idea of this result, we have recently proven that under some certain conditions of \((g,\tau,\sigma)\), there exists a sequence \(\{t_{n}\}\) converging to \(0\) such that the conformal equations associated to \((g,t_{n}\tau,\sigma)\) has at least two solutions. In this short talk, we would like to introduce the half-continuity method applied to the vacuum Einstein conformal constraint equations. More precisely, by using this method, we show that the result of Dahl-Gicquaud-Humbert is still valid for the vanishing \(\tau\), and nonuniqueness results above can be extended to all \(t\) small enough.Non UBCUnreviewedAuthor affiliation: Universite Francois-Rabelais de ToursGraduat