On the Cauchy Problem for Energy Critical Self-Gravitating Wave Maps

Abstract

This work is on the Cauchy problem for critical wave maps coupled to Einstein's equations of general relativity. The main result of this work is the proof that the energy of the Einstein-equivariant wave map system does not concentrate during the Cauchy evolution. A key ingredient in the proof is the use of the fact that geometric mass at infinity of the Einstein-equivariant wave map system is conserved during the evolution. However, this observation has some subtle local implications which have been used to estimate the energy locally. For instance, we construct a divergence-free vector field which gives monotonicity of energy in the past null cone of any point. In addition, this vector has also been used to prove that the energy does not concentrate away from the axis of the domain manifold. Later, estimating the divergence of a Morawetz vector on a truncated past null cone, we prove that the kinetic energy does not concentrate. Finally, assuming that the target manifold satisfies the Grillakis condition, we proceed to prove the non-concentration of energy for the critical Einstein-equivariant wave map system. Keeping track of various quantities of wave map relative to the evolving null geometry of the background manifold is a recurring theme throughout the course of this work. Apart from a purely mathematical interest, the motivation to study critical self-gravitating wave maps is that they occur naturally in 3+1 Einstein's equations of general relativity. Therefore, studying critical self-gravitating wave maps could be a fruitful way of understanding the ever elusive global behaviour of Einstein's equations. This work is a step in this endeavour.