Hermitian Curvature Flow on Compact Homogeneous Spaces

Panelli, Francesco; Podestà, Fabio

doi:10.1007/s12220-019-00239-7

Cited by 9 publications

(5 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For other relevant results on the Anomaly flow, we refer the reader to [4,12,14,37,39]; while, for some results on the Hermitian curvature flow in the homogeneous setting we refer the reader to [3,10,28,36,35,42,43,46,50].…”

Section: Introductionmentioning

confidence: 99%

The Hull-Strominger system and the Anomaly flow on a class of solvmanifolds

Pujia

2021

Preprint

View full text Add to dashboard Cite

We study the Hull-Strominger system and the Anomaly flow on a special class of 2-step solvmanifolds, namely the class of almost-abelian Lie groups. In this setting, we characterize the existence of invariant solutions to the Hull-Strominger system with respect to the family of Gauduchon connections in the anomaly cancellation equation. Then, motivated by the results on the Anomaly flow, we investigate the flow of invariant metrics in our setting, proving that it always reduces to a flow of a special form. Finally, under an extra assumption on the initial metrics, we show that the flow is immortal and, when the slope parameter is zero, it always converges to a Kähler metric

show abstract

Section: Introductionmentioning

confidence: 99%

The Hull-Strominger system and the Anomaly flow on a class of solvmanifolds

Pujia

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Several members of the HCF family are actively being studied in the homogeneous case. We refer the reader to [1,4,7,8,12,18,19,20,21,22,24].…”

Section: Introductionmentioning

confidence: 99%

Positive Hermitian Curvature Flow on special linear groups and perfect solitons

Stanfield¹

2021

Preprint

View full text Add to dashboard Cite

We study invariant solutions to the Positive Hermitian Curvature Flow, introduced by Ustinovskiy, on complex Lie groups. We show in particular that the canonical scale-static metrics on the special linear groups, arising from the Killing form, are dynamically unstable. This disproves a conjecture of Ustinovskiy. We also construct certain perfect Lie groups that admit at least two distinct invariant solitons for the flow, only one of which is algebraic. This is the second known example of a geometric flow with non-algebraic, homogeneous solitons. The first being the G2-Laplacian flow.

show abstract

“…In [22], Ustinovskiy studied the HCF + on complex homogeneous manifolds G/H. He showed that the space of (generically not G-invariant) metrics induced by left-invariant metrics on G is preserved, and that the flow in this case is governed by an ODE on the Lie algebra of G which is independent of the isotropy H. In [15], Panelli and Podestà studied the HCF + on compact homogeneous spaces. And the HCF + on 2-step nilpotent Lie groups was studied by Pujia in [18].…”

Section: Introductionmentioning

confidence: 99%

Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups

Stanfield

2020

Preprint

View full text Add to dashboard Cite

We study the positive Hermitian curvature flow on the space of left-invariant metrics on complex Lie groups. We show that in the nilpotent case, the flow exists for all positive times and subconverges in the Cheeger-Gromov sense to a soliton. We also show convergence to a soliton when the complex Lie group is almost abelian. That is, when its Lie algebra admits a (complex) co-dimension one abelian ideal. Finally, we study solitons in the almost-abelian setting. We prove uniqueness and completely classify all left-invariant, almost-abelian solitons, giving a method to construct examples in arbitrary dimensions, many of which admit co-compact lattices. Contents 1. Introduction 1 2. Preliminaries 4 2.1. HCF + on complex Lie groups 4 2.2. The bracket flow 5 2.3. Static metrics and solitons 7 3. HCF + on complex nilpotent Lie groups 9 3.1. Nilpotent HCF + solitons 3.2. Long time behaviour of the HCF + on nilpotent Lie groups 4. HCF + on almost-abelian complex Lie groups 4.1. Almost-abelian complex Lie groups 4.2. The behaviour of HCF + on almost-abelian Lie groups 5. Almost-abelian HCF + solitons 5.1. Classification of algebraic solitons References

show abstract

Hermitian Curvature Flow on Compact Homogeneous Spaces

Cited by 9 publications

References 21 publications

The Hull-Strominger system and the Anomaly flow on a class of solvmanifolds

The Hull-Strominger system and the Anomaly flow on a class of solvmanifolds

Positive Hermitian Curvature Flow on special linear groups and perfect solitons

Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups

Contact Info

Product

Resources

About