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Cynthia E. Will

Publications and source records attributed to Cynthia E. Will.

15 recordsLinked to original sources
arXiv2023

Bismut Ricci flat generalized metrics on compact homogeneous spaces (including a Corrigendum)

A generalized metric on a manifold $M$, i.e., a pair $(g,H)$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, is a fixed point of the generalized Ricci flow if and only if $(g,H)$ is Bismut Ricci flat: $H$ is $g$-harmonic and $ric(g)=\tfrac{1}{4} H_g^2$. On any homogeneous space $M=G/K$, where $G=G_1\times G_2$ is a compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit a Bismut Ricci flat $G$-invariant generalized metric, which is proved to be unique among a $4$-parameter space of metrics in many cases, including when $K$ is neither abelian nor semisimple. On the other hand, if $K$ is simple and the standard metric is Einstein on both $G_1/π_1(K)$ and $G_2/π_2(K)$, we give a one-parameter family of Bismut Ricci flat $G$-invariant generalized metrics on $G/K$ and show that it is most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for every space of the form $M=G\times G/ΔK$ and for $M^{35}=SO(8)\times SO(7)/G_2$. A Corrigendum has been added in Appendix A.

math.DG↗
arXiv2023

Harmonic 3-forms on compact homogeneous spaces

The third real de Rham cohomology of compact homogeneous spaces is studied. Given $M=G/K$ with $G$ compact semisimple, we first show that each bi-invariant symmetric bilinear form $Q$ on $\mathfrak{g}$ such that $Q|_{\mathfrak{k}\times\mathfrak{k}}=0$ naturally defines a $G$-invariant closed $3$-form $H_Q$ on $M$, which plays the role of the so called Cartan $3$-form $Q([\cdot,\cdot],\cdot)$ on the compact Lie group $G$. Indeed, every class in $H^3(G/K)$ has a unique representative $H_Q$. Secondly, focusing on the class of homogeneous spaces with the richest third cohomology (other than Lie groups), i.e., $b_3(G/K)=s-1$ if $G$ has $s$ simple factors, we give the conditions to be fulfilled by $Q$ and a given $G$-invariant metric $g$ in order for $H_Q$ to be $g$-harmonic, in terms of algebraic invariants of $G/K$. As an application, we obtain that any $3$-form $H_Q$ is harmonic with respect to the standard metric, although for any other normal metric, there is only one $H_Q$ up to scaling which is harmonic. Furthermore, among a suitable $(2s-1)$-parameter family of $G$-invariant metrics, we prove that the same behavior occurs if $\mathfrak{k}$ is abelian: either every $H_Q$ is $g$-harmonic (this family of metrics depends on $s$ parameters) or there is a unique $g$-harmonic $3$-form $H_Q$ (up to scaling). In the case when $\mathfrak{k}$ is not abelian, the special metrics for which every $H_Q$ is $g$-harmonic depend on $3$ parameters.

math.DG↗
arXiv2022

On the stability of homogeneous Einstein manifolds II

For any $G$-invariant metric on a compact homogeneous space $M=G/K$, we give a formula for the Lichnerowicz Laplacian restricted to the space of all $G$-invariant symmetric $2$-tensors in terms of the structural constants of $G/K$. As an application, we compute the $G$-invariant spectrum of the Lichnerowicz Laplacian for all the Einstein metrics on most generalized Wallach spaces and any flag manifold with $b_2(M)=1$. This allows to deduce the $G$-stability and critical point types of each of such Einstein metrics as a critical point of the scalar curvature functional.

math.DG↗
arXiv2021

Prescribing Ricci curvature on homogeneous spaces

The prescribed Ricci curvature problem in the context of G-invariant metrics on a homogeneous space M=G/K is studied. We focus on the metrics at which the Ricci curvature map is, locally, as injective and surjective as it can be. Our main result is that such property is generic in the compact case. Our main tool is a formula for the Lichnerowicz Laplacian we prove in terms of the moment map for the variety of algebras.

math.DG↗
arXiv2019

On Ricci negative Lie groups

We give an overview of what is known on Lie groups admitting a left-invariant metric of negative Ricci curvature, including many natural questions and conjectures in the solvable case. We also introduce an open and convex cone C(n) of derivations attached to each nilpotent Lie algebra n, which is defined as the image of certain moment map and parametrizes a set of solvable Lie algebras with nilradical n admitting Ricci negative metrics.

math.DG↗
arXiv2019

Non-solvable Lie groups with negative Ricci curvature

Until a couple of years ago, the only known examples of Lie groups admitting left-invariant metrics with negative Ricci curvature were either solvable or semisimple. We use a general construction from a previous article of the second named author to produce a great amount of examples with compact Levy factor. Given a compact semisimple real Lie algebra $\mathfrak u$ and a real representation $π$ satisfying some technical properties, the construction returns a metric Lie algebra $\mathfrak l(\mathfrak u,π)$ with negative Ricci operator. In this paper, when $\mathfrak u$ is assumed to be simple, we prove that $\mathfrak l(\mathfrak u,π)$ admits a metric having negative Ricci curvature for all but finitely many finite-dimensional irreducible representations of $\mathfrak u\otimes_{\mathbb R} \mathbb C$, regarded as a real representation of $\mathfrak u$. We also prove in the last section a more general result where the nilradical is not abelian, as it is in every $\mathfrak l(\mathfrak u,π)$.

math.DG↗
arXiv2019

The Ricci pinching functional on solvmanifolds II

It is natural to ask whether solvsolitons are global maxima for the Ricci pinching functional F:=scal^2/|Ric|^2 on the set of all left-invariant metrics on a given solvable Lie group S, as it is to ask whether they are the only global maxima. A positive answer to both questions was given in a recent paper by the same authors when the Lie algebra s of S is either unimodular or has a codimension-one abelian ideal. In the present paper, we prove that this also holds in the following two more general cases: 1) s has a nilradical of codimension-one; 2) the nilradical n of s is abelian and the functional F is restricted to the set of metrics such that a is orthogonal to n, where a is the orthogonal complement of n with respect to the solvsoliton.

math.DG↗
arXiv2019

Negative Ricci curvature on some non-solvable Lie groups II

We construct many examples of Lie groups with compact Levi factor admitting a left-invariant metric with negative Ricci curvature. We start with a Lie algebra with Levi factor su(n) or so(n) acting on an abelian nilradical via the representation on the space of homogeneous polynomials. In the case of su(2) we obtain a more general construction where the nilradical can be any nilpotent Lie algebra. We also prove a general result in the case when the Levi factor is a semisimple Lie algebra of non-compact type.

math.DG↗
arXiv2019

The Ricci pinching functional on solvmanifolds

We study the natural functional F=scal^2/|Ric|^2 on the space of all non-flat left-invariant metrics on all solvable Lie groups of a given dimension n. As an application of properties of the beta operator, we obtain that solvsolitons are the only global maxima of F restricted to the set of all left-invariant metrics on a given unimodular solvable Lie group, and beyond the unimodular case, we obtain the same result for almost-abelian Lie groups. Many other aspects of the behavior of F are clarified.

math.DG↗
arXiv2015

Characterization of 9-dimensional Anosov Lie algebras

The classification of all real and rational Anosov Lie algebras up to dimension 8 is given by Lauret and Will. In this paper we study 9-dimensional Anosov Lie algebras by using the properties of very special algebraic numbers and Lie algebra classification tools. We prove that there exists a unique (up to a Lie algebra isomorphism) complex 3-step Anosov Lie algebra of dimension 9. In the 2-step case, we prove that a 2-step real 9-dimensional Anosov Lie algebra with no abelian factor must have a 3-dimensional derived algebra and we characterize these Lie algebras in terms of their Pfaffian forms. Among these Lie algebras, we have found a family of infinitely many complex nonisomorphic Anosov Lie algebras.

math.DS↗
arXiv2014

Anosov automorphisms on nilmanifolds in dimensions 9 and 10

We study 9 and 10-dimensional Anosov Lie algebras, by using the properties of very special algebraic numbers. We classify k-step complex Anosov Lie algebras for $k>2$ and in the two step case, we give an example in each possible type, or we give a non-existence result.

math.DS↗
arXiv2008

A curve of nilpotent Lie algebras which are not Einstein nilradicals

The only known examples of noncompact Einstein homogeneous spaces are standard solvmanifolds (special solvable Lie groups endowed with a left invariant metric), and according to a long standing conjecture, they might be all. The classification of Einstein solvmanifolds is equivalent to the one of Einstein nilradicals, i.e. nilpotent Lie algebras which are nilradicals of the Lie algebras of Einstein solvmanifolds. Up to now, there have been found very few examples of graded nilpotent Lie algebras that can not be Einstein nilradicals. In particular, in each dimension, there are only finitely many known. We exhibit in the present paper two curves of pairwise non-isomorphic 9-dimensional 2-step nilpotent Lie algebras which are not Einstein nilradicals.

math.DG↗
arXiv2006

Examples of Anosov Lie Algebras

We construct new families of examples of (real) Anosov Lie algebras starting with algebraic units. We also give examples of indecomposable Anosov Lie algebras (not a direct sum of proper Lie ideals) of dimension 13 and 16, and we conclude that for every $n \geq 6$ with $n \neq 7$ there exists an indecomposable Anosov Lie algebra of dimension $n$.

math.DS↗
arXiv2004

Anosov diffeomorphisms on nilmanifolds up to dimension 8

After more than thirty years, the only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds. It is also important to note that the existence of an Anosov automorphism is a really strong condition on an infranilmanifold. Any Anosov automorphism determines an automorphism of the (rational) Lie algebra of the Mal'cev completion of the corresponding lattice which is hyperbolic and unimodular. These two conditions together are strong enough to make of such rational nilpotent Lie algebras (called Anosov Lie algebras) very distinguished objects. In this paper, we classify Anosov Lie algebras of dimension less or equal than 8, which also classify nilmanifolds admitting an Anosov diffeomorphism in those dimensions. As a corollary we obtain that if an infranilmanifold of dimension n<9 admits an Anosov diffeomorphism f and it is not a torus or a compact flat manifold (i.e. covered by a torus), then n=6 or 8 and the signature of f necessarily equals {3,3} or {4,4}, respectively. We had to study the set of all rational forms up to isomorphism for many real Lie algebras, which is a subject on its own and it is treated in a section completely independent of the rest of the paper.

math.DS↗