On the Existence and Stability of Filiform Nilsolitons

Abstract

This dissertation is concerned with the existence and stability of nilsoliton metrics on filiform Lie algebras. The results are presented in two major components. First, we give new results which preclude the existence of soliton metrics on rank 1 filiform algebras. Most notably, these methods circumvent the need to classify the algebras in each dimension (a major obstacle to the study, to this point). Second, we demonstrate that all soliton metrics on rank 2 filiform algebras are stable. In the course of this, we will develop new approximation techniques, and compute the full curvature tensor of rank 2 filiform algebras. The tables for these curvature tensors are contained in Appendix A.