The long‐time behavior of the homogeneous pluriclosed flow

Abstract

We study the asymptotic behavior of the pluriclosed flow in the case of left‐invariant Hermitian structures on Lie groups. We prove that solutions on 2‐step nilpotent Lie groups and on almost‐abelian Lie groups converge, after a suitable normalization, to self‐similar solutions of the flow. Given that the spaces are solvmanifolds, an unexpected feature is that some of the limits are shrinking solitons. We also exhibit the first example of a homogeneous manifold on which a geometric flow has some solutions with finite extinction time and some that exist for all positive times.

Hermitian curvature flow on unimodular Lie groups and static invariant metrics

“…In [7], Boling studied locally homogeneous solutions to the PCF on compact complex surfaces, and in [31] Lauret gives a regularity result for a class of flows on Lie groups which includes PCF. Also, an analogous result to Theorem 1.1 for the PCF on 2-step nilmanifolds was recently obtained in [4].…”

“…The bracket flow approach. In this section we recall the bracket flow device introduced by Lauret in [28] and used in [4,17,31,32,33] to study geometric flows of (almost)-Hermitian structures. The trick consists in regarding the flow as an evolution equation for Lie brackets instead of metrics.…”

“…see also [4,Corollary 3.5] and [28]. By compactness, the family of unit norm brackets (ν t ) t∈[0,∞) must have an accumulation pointν.…”

Finding Solitons

“…The space Γ is now given by , , the vector space of all homogeneous polynomials of degree in variables with coefficients in ℝ (e.g., quadratic ( = 2) and binary ( = 2) forms). There is a natural left GL -action on , given by ℎ ⋅ ≔ ∘ ℎ −1 and the inner product for which the basis of monomials { ≔ 1 1 ⋯…”

Hermitian Curvature Flow on Compact Homogeneous Spaces

“…While the analysis of a generic HCF on these manifolds seems to be out of reach, some special flows may deserve attention. In view of the classification results obtained in [5] for C-spaces carrying invariant SKT metrics, the pluriclosed flow can be investigated only in very particular cases of such spaces (see also [3] for the analysis of the pluriclosed flow on compact locally homogeneous surfaces and [2] for the case of left-invariant metrics on Lie groups). On the other hand the HCF U is geometrically meaningful and can be dealt with more easily.…”