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

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Relativity and gravitational theory; Differential geometry

video/mp4

Author affiliation: Universite Francois-Rabelais de Tours

2017-02-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60362

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/