On Singular Liouville Equations and Systems

We consider some singular Liouville equations and systems motivated by uniformization problems in a non-smooth setting, as well as from models in mathematical physics. We will study the existence of solutions from a variational point of view, using suitable improvements of the Moser–Trudinger inequality. These reduce the problem to a topological one by studying the concentration property of conformal volume, which will be constrained by the functional inequalities of geometric flavour. We will mainly describe some common strategies from the papers [

Spectre et géométrie conforme des variétés compactes à bord

We prove that on any compact manifold $M^{n}$ with boundary, there exists a conformal class $C$ such that for any Riemannian metric $g\in C$ of unit volume, the first positive eigenvalue of the Neumann Laplacian satisfies ${\it\lambda}_{1}(M^{n},g)<n\,\text{Vol}(S^{n},g_{\text{can}})^{2/n}$. We also prove a similar inequality for the first positive Steklov eigenvalue. The proof relies on a handle decomposition of the manifold. We also prove that the conformal volume of $(M,C)$ is $\text{Vol}(S^{n},g_{\text{can}})$, and that the Friedlander–Nadirashvili invariant and the Möbius volume of $M$ are equal to those of the sphere. If $M$ is a domain in a space form, $C$ is the conformal class of the canonical metric.

THE PHASE PORTRAIT OF THE REDUCED EINSTEIN EQUATIONS

We discuss the phase portrait of the reduced Einstein (3+1)-equations on a compact manifold of Yamabe type -1. We show that the flow for these equations either has a unique fixed point if the underlying manifold M is hyperbolizable or has no fixed points if M is not hyperbolizable. Thus the topology of M is a critical determinant of the phase portrait of the reduced equations. If, additionally, M is rigid, the fixed point is a local attractor, thereby answering an important question regarding the stability of these model universes. In the non-hyperbolizable case, under certain conditions, the reduced Einstein flow predicts that the conformal volume collapses M, in contrast to the physical volume which, as befits an expanding universe, goes to infinity as the coordinate time goes to infinity.