The geometry of lorentzian ricci solitons

Resumo

The investigation of rigidity phenomena is a central and wide topic in pseudo-Riemannian geometry. Rigidity results may appear at the metric level, like splitting theorems, or at the topological level, being compactness theorems or results involving the first fundamental group classical examples. Moreover, if the manifold is equipped with some additional structure, one analyzes its behavior as it often gives rise to restrictions at both levels. In this thesis we consider Lorentzian manifolds equipped with an additional structure given by certain differential equations: the Ricci soliton and the quasi-Einstein equations. Traditionally, in Analysis, one is mostly interested in the existence of a nontrivial solution to a differential equation on a certain domain. However, from a more geometric point of view, one can also argue the existence of a domain manifold or structure for a differential equation to provide a nontrivial solution, and this leads to a rigidity result for the corresponding structure. Ricci solitons and quasi-Einstein metrics can be viewed as generalizations of Einstein manifolds. However, the motivation for their study comes from different problems. The ultimate aim of the different geometric evolution equations is to produce (or deduce the existence of) manifolds with an optimal behavior with respect to the given invariants: the Ricci flow makes it possible to construct Einstein metrics under certain conditions, whereas the mean curvature flow makes it possible to deform certain surfaces in other ones whose mean curvature is constant. However, there are conditions under which the initial structure does not evolve under the flow but remains as a fixed point of it. Ricci solitons are the geometric fixed points (modulo homotheties and diffeomorphisms) of the Ricci flow. Moreover, since they appear as singular models for the flow, analyzing their geometry is an important step towards an understanding of the Ricci flow itself. The quasi-Einstein equation appears as a natural equation associated to the Lichnerowicz-Bakry-Emery Ricci tensor, which is a modified Ricci curvature on weighted manifolds. Moreover, the quasi-Einstein equation encodes necessary and sufficient information to construct Einstein warped product metrics. It has been our purpose in this work to investigate Ricci solitons and quasi-Einstein metrics under some natural curvature conditions and show some rigidity phenomena for both structures. Within the family of Ricci solitons, we have paid special attention to gradient Ricci solitons. Essentially, both conditions, namely gradient Ricci solitons and quasi-Einstein equations, provide information about the level sets of the corresponding potential function and the Ricci curvature of the underlying manifold. Hence, one naturally focuses on locally conformally flat spaces, as in this case the Ricci tensor determines the curvature and one can show a local rigidity for the structures under consideration. It is worth emphasizing here that, in general, working in the Lorentzian setting is less rigid than in its Riemannian analog, since Lorentzian geometry allows degenerate hypersurfaces which may occur as level sets of the solutions of the differential equations under consideration. Special attention is also paid to the existence of solutions for the Ricci soliton equation in manifolds with a certain degree of homogeneity. Our results are especially conclusive in the three-dimensional context, where we provide a complete description of homogeneous gradient Ricci solitons. Also, for manifolds with large isometry group, we show that they all support expanding, steady and shrinking Ricci solitons.