Regularity results for pluriclosed flow

Abstract

In [8] the authors introduced a parabolic flow of pluriclosed metrics. New advancements in this flow are given, including improved regularity results, a gradient property and expanding entropy functional, and a conjectural picture of optimal existence results and their consequences. Let (M2n, J) be a complex manifold, and let ω denote a Hermitian metric on M . The metric ω is pluriclosed if ∂∂ω = 0. Consider the initial value problem ∂ ∂t ω = ∂∂∗ω + ∂∂∗ω + √−1 2 ∂∂ log det g ω(0) = ω0. (1) This equation was introduced in [8] as a tool for understanding complex, non-Kahler manifolds. Equation (1) falls into a general class of flows of Hermitian metrics, and as shown in [7], solutions to (1) exist as long as the Chern curvature, torsion, and covariant derivative of torsion are bounded. In this note we will describe many results which refine the existence and singularity formation of solutions to (1), which appear in [9]. The starting point is to interpret equation (1) using the Bismut connection. This is a small but crucial observation which clarifies many aspects of (1). Let us recall the Bismut connection. Let (M2n, ω, J) be a complex manifold with pluriclosed metric, and let D denote the Levi Civita connection. Then the Bismut connection ∇ is defined by 〈∇XY, Z〉 = 〈DXY, Z〉+ 12d ω(X, Y, Z) where dcω(X,Y, Z) := dω(JX, JY, JZ). Let Ω denote the curvature of this connection, and let P denote the Chern form of this connection, i.e. in complex coordinates Pij = Ω k ijk Finally, let PC denote the Ricci form associated to the Chern connection. One can calculate that P = P − dd∗ω. In particular, this implies that a solution to (1) may be expressed as ∂ ∂t ω = − P 1,1 (2) where P 1,1 denotes the projection of P onto (1, 1)-forms. There is another geometric evolution equation naturally associated to a connection with torsion given by a closed Date: August 16, 2010. 1 2 JEFFREY STREETS AND GANG TIAN three-form, namely the B-field renormalization group flow of string theory. A remarkable fact is that (2) is equivalent to this B-field flow, after a change by diffeomorphism. Thus Perelman-type monotone quantities which had previously been discovered for the B-field flow, specifically an energy functional [4] and expanding entropy functional [6], carry over to solutions to (1). To describe the energy functional, first generalize the notation slightly and let (Mn, g) be a Riemannian manifold, and let T denote a three-form on M . Let