## Workshop on Ricci Solitons and Symmetry

• March 16-18, 2012

# Abstracts

Abstracts of all lectures are needed for our records. Several participants have also expressed interest in having them available by the beginning of the meeting. Please email us your abstract as soon as possible, not to arrive later than March 9.

Charles Boyer, University of New Mexico
Extremal Sasakian metrics

I begin the talk by giving a brief general discussion about Sasakian geometry and extremal structures in particular. I want to focus on several foundational problems: 1. Given a manifold determine how many contact structures of Sasaki type there are. 2. Given an isotopy class of contact structures determine the space of compatible extremal Sasakian structures. 3. Given extremal Sasakian structures when do they have constant scalar curvature? We concentrate our discussion on certain examples in dimension five.

Maria Buzano, University of Oxford

In this talk we are going to study the gradient Ricci soliton equation in the cohomogeneity one setting. We will begin by deriving the system of nonlinear ODEs which corresponds to the equation in this case. Then, we will study the intial value problem around a singular orbit of the cohomogeneity one manifold and present a generalisation of a previous result by Eschenburg and Wang for the Einstein case.

Edison Fernández-Culma, Universidad Nacional de Córdoba
The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety $\mathfrak{N}_n(\mathbb C)$ of $n$-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which $\mathrm{GL}_n(\mathbb C)$-orbits in $\mathfrak{N}_n(\mathbb C)$ have a critical point of the squared norm of the moment map. We have proved in arXiv:1105.4489 a classification theorem of such distinguished orbits for $n = 7$. The set $\mathfrak{N}_7(\mathbb C) / \mathrm{GL}_7(\mathbb C)$ is formed by $148$ nilpotent Lie algebras and $6$ one-parameter families of pairwise non-isomorphic nilpotent Lie algebras. We have applied to each Lie algebra one of three main techniques (based on the earlier results of J. Lauret and Y. Nikolayevsky) to decide whether it has a distinguished orbit or not. In the talk, we will give many examples which cover all possible strategies that we have used to obtain the full classification; each Lie algebra in the work of Magnin (Adjoint and Trivial Cohomology Tables for Indecomposable Nilpotent Lie Algebras of Dimension ≤ 7 over $\mathbb C$, 2007) has been worked out in some of these ways.

Eduardo Garcia-Rio, University of Santiago de Compostela
Lorentzian Quasi-Einstein metrics

Quasi-Einstein metrics are considered in a Lorentzian setting. After emphasizing its relation with gradient Ricci solitons and warped product Einstein spaces, our main purpose is to understand the underlying geometry of quasi-Einstein metrics under some curvature condition. We focus on locally conformally flat metrics, showing the existence of some Lorentzian examples without Riemannian analog. This motivates a detailed study of plane-waves, investigating some differences between quasi-Einstein metrics and gradient Ricci solitons.

Brett Kotschwar, Arizona State University
Unique continuation, with and without analyticity, for the Ricci flow

Certain questions arising in the study of the Ricci flow can be cast as problems of unique continuation, among them, whether the isometry and holonomy groups of solutions remain unchanged along the flow, and whether a solution can become self-similar or Einstein spontaneously within its smooth lifetime. I will discuss two approaches to these problems -- one an embedding of the problem into a larger system of mixed partial and ordinary differential inequalities amenable to Carleman-type estimates, and the second, more recent approach based on establishing estimates which imply the (interior) time-analyticity of the flow.

Jorge Lauret, Universidad Nacional de Córdoba
On homogeneous Ricci solitons

We shall give an overview on the classification of Ricci soliton Riemannian metrics which are homogeneous, including nilsolitons, solvsolitons, algebraic solitons and semi-algebraic solitons.

Tanya Lloyd-Hepburn, Saint Louis University
Ricci Flow on Some Classes of Naturally Reductive Homogeneous Spaces
We will examine Ricci flow in some classes of Naturally Reductive Homogeneous Spaces. In particular we want to relate Ricci Flow on a naturally reductive homogeneous space $M=G/H$, to Ricc Flow on $G$. We will show that $Ric^M=2Ric^G|_{\mathfrak{m}\times\mathfrak{m}}$ under certain conditions.

Kensuke Onda, Nagoya University
Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups
In this talk, we study Algebraic Ricci Solitons in Lorentzian case, and classify Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are solvsolitons. In particular, we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not to be algebraic Ricci solitons.

Tracy Payne, Idaho State University
Geometric Invariants for Nilpotent Metric Lie Algebras with Applications to Moduli Spaces of Nilsoliton Metrics

We give a survey of geometric and algebraic invariants for nilpotent metric Lie algebras. We present new invariants that can be used to determine whether two nilpotent metric Lie algebras are in the same isometry class. We apply these invariants to present explicit examples of large continuous families of nilsoliton metric Lie algebras that are deformations of uniform metric Lie algebras.

Peter Petersen, University of California, Los Angeles
Warped Product Rigidity

The goal is to give a PDE characterization of when a Riemannian manifold is a warped product with constant curvature fibers. This problem is motivated by the study of warped product Einstein metrics, where it has been noted that some spaces allow for many inequivalent warped product structures, while others only have essentially one such structure. The results will also be used in Will Wylie's talk.

Cynthia Will, Universidad Nacional de Córdoba
The space of Solvsolitons in low dimensions
Up to now, the only known examples of (nontrivial) homogeneous Ricci soliton metrics are the so called solvsolitons; namely, certain left invariant metrics on simply connected solvable Lie groups. In this talk, we will describe the ideas we used to classify the moduli space of solvsolitons of dimension less or equal than 7, up to isometry and scaling. We start with the already known classification of nilsolitons and, by following a characterization given by Lauret, we describe the subspace of solvsolitons associated to a given nilsoliton as the quotient of a Grassmanian by a finite group.

Mike Williams, University of California, Los Angeles
Solitons derived from Heisenberg groups
Lauret recently characterized solvsolitons as certain extensions of nilsolitons. Will subsequently described the moduli space of any such extension as the quotient of a Grassmannian by a finite group. We illustrate these results in the case of nilsolitons on Heisenberg groups of arbitrary odd dimension.

Will Wylie, Syracuse University
Ricci solitons and warped product Einstein metrics

I'll show how every algebraic soliton on a simply connected Lie group can be extended to a warped product Einstein metric. This comes thru studying the m-Quasi Einstein equation. I'll also discuss other connections between these equations..