[Paper Review] Regularity results for pluriclosed flow

Jeff Streets, Gang Tian|arXiv (Cornell University)|Aug 16, 2010

Geometric Analysis and Curvature Flows16 references15 citations

TL;DR

This paper establishes improved long-time existence and regularity results for the pluriclosed flow on non-Kähler complex manifolds by identifying it as a gradient flow of the lowest eigenvalue of a Schrödinger operator via the Bismut connection. It proves that bounded Bismut Ricci curvature implies smooth extension past singularities, and shows the flow arises as the B-field renormalization group flow in string theory, yielding an expanding entropy functional and implying breather solutions are gradient solitons.

ABSTRACT

In prior work the authors introduced a parabolic flow of pluriclosed metrics. Here we give improved regularity results for solutions to this equation. Furthermore, we exhibit this equation as the gradient flow of the lowest eigenvalue of a certain Schrödinger operator, and show the existence of an expanding entropy functional for this flow. Finally, we motivate a conjectural picture of the optimal regularity results for this flow, and discuss some of the consequences.

Motivation & Objective

• To establish sharper long-time existence theorems for the pluriclosed flow on complex manifolds with non-Kähler metrics.
• To show that the pluriclosed flow is the gradient flow of the lowest eigenvalue of a Schrödinger operator, using the Bismut connection.
• To exhibit an expanding entropy functional for the flow, implying monotonicity and strong long-time behavior.
• To motivate a conjectural picture of optimal regularity and singularity formation, with implications for the topology of Class VII+ surfaces.
• To connect the flow to mathematical physics by identifying it as the B-field renormalization group flow in nonlinear sigma models.

Proposed method

• The authors use the Bismut connection as a natural geometric framework to analyze the pluriclosed flow, replacing the Chern and Levi-Civita connections.
• They express the flow as the gradient flow of the functional λ(g,T), defined as the infimum of a weighted energy functional over normalized functions f.
• The Bismut connection curvature P is used to rewrite the flow as ∂ₜω = −P¹¹, where P¹¹ is the (1,1)-part of the Bismut curvature.
• They derive an expanding entropy functional for the flow, analogous to Perelman’s entropy in Ricci flow, which is monotone along solutions.
• By analyzing the evolution of the Bismut Ricci curvature, they prove that bounded L¹ norm of |P¹¹| over time implies smooth extension past singularities.
• They apply the maximum principle to the Bochner formula for holomorphic forms to derive vanishing theorems for static metrics.

Experimental results

Research questions

• RQ1 Does the pluriclosed flow admit improved long-time existence criteria that do not require bounded torsion and its covariant derivative?
• RQ2 Can the pluriclosed flow be interpreted as a gradient flow of a geometric functional, and if so, which one?
• RQ3 Does the flow admit a monotone entropy functional analogous to Perelman’s in Ricci flow?
• RQ4 What is the relationship between the pluriclosed flow and renormalization group flow in string theory with a B-field?
• RQ5 What are the topological and geometric consequences of the flow’s behavior on Class VII+ surfaces?

Key findings

• The pluriclosed flow is shown to be the gradient flow of the lowest eigenvalue λ(g,T) of a Schrödinger operator on the space of metrics and closed 3-forms modulo diffeomorphisms.
• An expanding entropy functional is constructed for the flow, which is monotone decreasing along solutions, implying strong long-time regularity properties.
• A solution to the pluriclosed flow extends smoothly past time τ if the integral of sup|P¹¹| over [0,τ) is finite, a result analogous to Sesum’s theorem for Ricci flow.
• Breather solutions of the flow are proven to be gradient solitons, a consequence of the entropy monotonicity and gradient flow structure.
• The flow is identified as the B-field renormalization group flow of a nonlinear sigma model, linking it to string theory and providing a physical interpretation.
• For compact complex manifolds with c₁ = 0 or c₁ > 0 in Aeppli cohomology, static solutions of the flow satisfy hⁿ⁻¹,⁰ = 0 unless the metric is Kähler, implying strong rigidity.