Regularity results for pluriclosed flow

Тиан

2012

Nuclear Physics B

Self Cite

Abstract. In [16] the authors introduced a parabolic flow for pluriclosed metrics, referred to as pluriclosed flow. We also demonstrated in [17] that this flow, after certain gauge transformations, gives a class of solutions to the renormalization group flow of the nonlinear sigma model with B-field. Using these transformations, we show that our pluriclosed flow preserves generalized Kähler structures in a natural way. Equivalently, when coupled with a nontrivial evolution equation for the two complex structures, the B-field renormalization group flow also preserves generalized Kähler structure. We emphasize that it is crucial to evolve the complex structures in the right way to establish this fact.

Section: Introductionmentioning

confidence: 76%

mentioning

confidence: 75%

“…Let g ± (t) be the Hermitian metric on M whose Kähler form is ω ± (t). As in Theorem 6.5 of [17], we set (1.3)…”

Section: Introductionmentioning

confidence: 99%

“…In view of results in [17], one can ask: Does (1.5) preserve generalized Kähler geometry? As it turns out, in the naive sense in which this question is usually asked, the answer is no.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Generalized Kähler geometry and the pluriclosed flow

Тиан

2012

Nuclear Physics B

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Classification of Generalized Kähler‐Ricci Solitons on Complex Surfaces

Ustinovskiy

2020

Comm Pure Appl Math

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Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in by the first author in 2019. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first examples of generalized Kähler structures with nonvanishing Poisson structure on nonstandard Hopf surfaces, completing the existence question for such structures. Moreover, this gives a complete answer to the existence question for generalized Kähler-Ricci solitons on compact complex surfaces. In the setting of generalized Kähler geometry with vanishing Poisson structure, we show that these solitons are unique. We show that these solitons are global attractors for the generalized Kähler-Ricci flow among metrics with maximal symmetry.

Pluriclosed Flow and the Geometrization of Complex Surfaces

2020

Geometric Analysis

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We recall fundamental aspects of the pluriclosed flow equation and survey various existence and convergence results, and the various analytic techniques used to establish them. Building on this, we formulate a precise conjectural description of the long time behavior of the flow on complex surfaces. This suggests an attendant geometrization conjecture which has implications for the topology of complex surfaces and the classification of generalized Kähler structures.Date: August 30, 2018.