Homogeneous Ricci solitons
Abstract
In this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat, it is necessarily simply-connected and diffeomorphic to ℝn.
In the general case, we prove that homogeneous Ricci solitons must be semi-algebraic Ricci solitons in the sense that they evolve under the Ricci flow by dilation and pullback by automorphisms of the isometry group. In the special case that there exists a transitive semi-simple group of isometries on a Ricci soliton, we show that such a space is in fact Einstein. In the compact case, we produce new proof that Ricci solitons are necessarily Einstein.
Lastly, we characterize solvable Lie groups which admit Ricci soliton metrics.
Funding source: NSF
Award Identifier / Grant number: DMS-1105647
It is my pleasure to thank Jorge Lauret and Dan Knopf for many helpful comments on an early version of this work.
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- Relatively very free curves and rational simple connectedness
- Moduli of unipotent representations II: Wide representations and the width
- Analog of selfduality in dimension nine
- A p-adic Eisenstein measure for unitary groups
- On the Kähler–Ricci flow near a Kähler–Einstein metric
- Homogeneous Ricci solitons
- Energy inequalities for cutoff functions and some applications
- Dimension of elementary amenable groups
Articles in the same Issue
- Frontmatter
- Relatively very free curves and rational simple connectedness
- Moduli of unipotent representations II: Wide representations and the width
- Analog of selfduality in dimension nine
- A p-adic Eisenstein measure for unitary groups
- On the Kähler–Ricci flow near a Kähler–Einstein metric
- Homogeneous Ricci solitons
- Energy inequalities for cutoff functions and some applications
- Dimension of elementary amenable groups
