ε-Regularity and Structure of Four-dimensional Shrinking Ricci Solitons

A closed four-dimensional manifold cannot possess a non-flat Ricci soliton metric with arbitrarily small $L^{2}$-norm of the curvature. In this paper, we localize this fact in the case of gradient shrinking Ricci solitons by proving an $\varepsilon $-regularity theorem, thus confirming a conjecture of Cheeger–Tian [20]. As applications, we will also derive structural results concerning the degeneration of the metrics on a family of complete non-compact four-dimensional gradient shrinking Ricci solitons without a uniform entropy lower bound. In the appendix, we provide a detailed account of the equivariant good chopping theorem when collapsing with locally bounded curvature happens.

Classification of Ordered Type Soliton Metric Lie Algebras by a Computational Approach

We classify some soliton nilpotent Lie algebras and possible candidates in dimensions 8 and 9 up toisomorphy.We focus on type of derivations, where is the dimension of the Lie algebras. We present algorithms to generate possible algebraic structures.