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Pluriclosed flow, Born-Infeld geometry, and rigidity results for generalized Kähler manifolds. (English) Zbl 1347.53055
Let $$(M^{2n},g,J)$$ be a Hermitian manifold. The metric is pluriclosed if $$\partial \bar{\partial} \omega=0$$. The pluriclosed flow is a natural geometric flow on complex manifolds which preserves pluriclosed metrics. The author proves long-time existence and convergence results for this flow, and geometric and topological rigidity results which follow as corollaries. A long-time existence and convergence result for the pluriclosed flow in the setting of commuting generalized Kähler geometry is also derived. A generalized Käher structure on a compact manifold $$M$$ is defined as a triple $$(g,J_A,J_B)$$ made of a Riemannian metric and two integrable complex structures satisfying the equations: $$d_A^c\omega_A=-d_B^c\omega_B$$, $$dd_A^c\omega_A=-dd_B^c\omega_B=0$$. To achieve these results the reduction of the pluriclosed flow to a degenerate parabolic equation for a $$(1,0)$$-form is used. The convergence of the flow is proved using a generalization of Yau’s oscillation estimate.
Moreover the author develops a number of new a priori estimates for the pluriclosed flow which lead to general long-time existence and convergence result, given some topological constraints. This implies a rigidity result showing that under certain topological conditions, generalized Kähler structures are automatically covered by products of Calabi-Yau manifolds. The first principal estimate is an a priori $$C^\alpha$$ estimate for the metric in the presence of upper and lower bounds on the metric. Considering of the pluriclosed flow as a parabolic system of equations for the Hermitian metric $$g$$, this estimate is analogous to the DeGiorgi-Nash-Moser/Krylov-Safonov estimate for uniformly parabolic equations. We point out that these are only analogies, and that moreover, the DeGiorgi-Nash-Moser/Krylov-Safonov results are known to be false in general for the systems of equations which are considered here. The relations among the methods exploited here and in other settings are also analyzed.
Anyhow, by a careful study of the 1-form potential $$\alpha$$ along the pluriclosed flow, the author discovers a judicious combination of the first derivatives of $$\alpha$$ into a Hermitian $$2n\times 2n$$ matrix $$W$$ such that $$W(v,v)$$ is a subsolution to a uniformly parabolic equation for every $$v$$, and such that $$\det W=1$$. This matrix $$W$$ admits an interpretation as a “Born-Infeld” metric on the generalized tangent bundle $$T\oplus T^\ast$$. This motivates the author to discuss the connection between the pluriclosed flow and generalized geometry. The second principle estimate is a general upper bound for the metric in terms of a lower bound. The proof exploits the very favorable evolution equations arising from the 1-form reduction of pluriclosed flow to control some torsion terms arising in the evaluation of metric quantities.

##### MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 53C55 Global differential geometry of Hermitian and Kählerian manifolds 35R01 PDEs on manifolds 35K96 Parabolic Monge-Ampère equations
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