Michael Jablonski

Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups

In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that one may determine the existence of a such a metric by analyzing algebraic properties of the Lie algebra and infinitesimal deformations of any initial metric. Our second main result concerns the isometry groups of such distinguished metrics. Among the completely solvable unimodular Lie groups (this includes nilpotent groups), if the Lie group admits such a metric, we show that the isometry group of this special metric is maximal among all isometry groups of left-invariant metrics.

Homogeneous Ricci solitons are algebraic

In this short note, we show that homogeneous Ricci solitons are algebraic. As an application, we see that the generalized Alekseevskii conjecture is equivalent to the Alekseevskii conjecture.

Distinguished orbits of reductive groups

We prove a generalization of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group

Strongly solvable spaces

G

Linear stability of algebraic Ricci solitons

acting linearly and rationally on a real vector space

Homogeneous Ricci solitons

V

Detecting orbits along subvarieties via the moment map

.

Moduli of Einstein and non-Einstein nilradicals

G

A step towards the Alekseevskii conjecture

can be viewed as the real points of a complex reductive group

Real Geometric Invariant Theory and Ricci soliton metrics on two-step nilmanifolds

G^\mathbb C