A flow of conformally balanced metrics with Kähler fixed points

Abstract

While the Anomaly flow was originally motivated by string theory, its zero slope case is potentially of considerable interest in non-Kähler geometry, as it is a flow of conformally balanced metrics whose stationary points are precisely Kähler metrics. We establish its convergence on Kähler manifolds for suitable initial data. We also discuss its relation to some current problems in complex geometry. Motivations for the flowWe provide now some details on the three different contexts which make the flow (1.1) of particular interest. The Hull-Strominger systemLet X be a compact 3-fold equipped with a nowhere vanishing holomorphic (3, 0)-form Ω. Let t → Φ(t) be a flow of (2, 2)-forms with dΦ = 0 for any t. Then the following flow was

On the existence of balanced metrics on six-manifolds of cohomogeneity one

“…A U(n)-structure on a 2n-dimensional smooth manifold M is the data of a Riemannian metric g and a g-orthogonal almost complex structure J . The pair (g, J ) is also known as an almost Hermitian structure on M. When J is integrable, i.e., (M, J ) is a complex manifold, the pair (g, J ) defines a Hermitian structure on M. In this case, the metric g is called balanced when dω n−1 = 0, ω:=g (J •, •) denoting the associated fundamental form, and we shall refer to (g, J ) as a balanced U(n)-structure on M. Balanced metrics have been extensively studied in [4,[10][11][12][13]23,25] (see also the references therein).…”

New curvature flows in complex geometry

“…In particular, from the above discussion, the Kähler property is a necessary condition for the convergence of the flow. At this moment, we can prove a partial converse [82]:…”

“…The special case α ′ = 0 of the Anomaly flow may actually be of independent geometric interest as well. It is given by ∂ t ( Ω ω ω 2 ) = i∂∂ω for 3-folds X, and a natural generalization to arbitrary n dimensions for n ≥ 3 can be defined by [82] ∂ t ( Ω ω ω n−1 ) = i∂∂ω n−2 . (6.1)…”

“…As in the case of 3 dimensions, the above flow of (n − 1, n − 1)-forms admits a useful reformulation as a flow of metrics, as it was given in [82]:…”

Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups

“…In [5], Fei and Phong showed that there exists a relation between the HCF + and the Anomaly flow, which is a metric flow introduced by Phong, Picard and Zhang in [15] to study the Hull-Storiminger system (see also [14,16,17]). Namely, Fei and Phong proved that any solution to the HCF + starting from a conformally balanced metric ω on X, i.e.…”