The pluriclosed flow preserves Vaisman condition on compact complex surfaces.
problem Preserving Vaisman condition under pluriclosed flow.
method Pluriclosed flow on compact complex surfaces.
result Preserves Vaisman condition if and only if starting metric has constant scalar curvature.
Study of pluriclosed flow on Oeljeklaus-Toma manifolds, showing convergence to a soliton.
problem Investigating the behavior of pluriclosed flow on Oeljeklaus-Toma manifolds.
method Parametrized left-invariant pluriclosed metrics, classified, and analyzed the flow's long-time behavior.
result The flow converges to an algebraic soliton, with normalized metrics collapsing to a torus.
Global existence and convergence of pluriclosed flow on Oeljeklaus-Toma manifolds.
problem Global existence and convergence of pluriclosed flow on specific complex manifolds.
method Established global existence with arbitrary initial data and Gromov-Hausdorff convergence of blowdown limits.
result Gromov-Hausdorff convergence of blowdown limits to a torus under conjectural bounds.
Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.
problem Preserving Hermitian-symplectic structures under pluriclosed flow.
method Consideration of an extra evolution equation determined by the Bismut-Ricci form.
result Obtained topological obstruction to long-time existence in arbitrary dimensions.
The study examines stability of specific geometric flows.
problem Stability of Pluriclosed and Generalized Ricci solitons.
method Analyzes the second variation of generalized Einstein--Hilbert functional and infinitesimal deformations.
result Stability of the flows and solitons under specific conditions.
Prove long-time existence of pluriclosed flow on certain fibrations
problem Long-time existence of pluriclosed flow on fibrations
method General theorem on holomorphic submersions
result Long-time existence of pluriclosed flow on nilmanifolds, almost-abelian solvmanifolds, and certain complex surfaces
Study on Vaisman metrics on Kodaira-Thurston surface using pluriclosed flow.
problem Characterizing and preserving Vaisman metrics on Kodaira-Thurston surface.
method Characterization of T2-invariant Vaisman metrics, analysis of pluriclosed flow behavior. result Pluriclosed flow preserves Vaisman condition on Kodaira-Thurston surface, including non-constant scalar curvature.
Derivative estimates for pluriclosed flow control curvature and torsion.
problem Deriving derivative estimates for the pluriclosed flow.
method Control higher order derivatives of Chern curvature and torsion using Chern curvature; derive an estimate for torsion tensor using Chern Ricci curvature in dimension two; find a monotonic quantity in Hermitian-symplectic case.
result All Hermitian-symplectic solitons are Kähler Ricci solitons.
Study describes global existence and convergence of flows on surfaces and fibrations.
problem Global existence and convergence of flows on surfaces and fibrations.
method Complete description of Ricci-Yang-Mills flow and pluriclosed flow on Tk bundles over Riemann surfaces. result Equivalence of solutions to generalized Ricci flow and pluriclosed flow with symmetry.
In prior work the authors introduced a parabolic flow for pluriclosed metrics, referred to as pluriclosed flow. We also demonstrated 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, w…
Study of t-Gauduchon Ricci-flat condition under Chern-Ricci flow on non-Kähler manifolds.
problem Investigating the t-Gauduchon Ricci-flat condition on non-Kähler manifolds. method Chern-Ricci flow approach, examples of non-Kähler Calabi-Yau manifolds, and geometric flow analysis.
result Examples of Chern-Ricci flow on non-Kähler Calabi-Yau manifolds that do not preserve the t-Gauduchon Ricci-flat condition. New insights prevent certain types of metrics on compact spaces.
problem Preventing the existence of specific types of metrics on compact spaces.
method Computing cohomology and analyzing stability of metrics for the pluriclosed flow.
result Prevents the existence of non-flat homogeneous Bismut Hermitian Einstein metrics on C-spaces.
Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time existence.
We show global existence and convergence results for the pluriclosed flow on manifolds for which certain naturally associated tensor bundles are globally generated.
Left-invariant metrics force 2-step nilpotent groups, preserving Kähler-like conditions.
problem Existence of left-invariant pluriclosed Hermitian metrics on Lie groups.
method Analyzing left-invariant metrics on unimodular Lie groups with abelian complex structures.
result Pluriclosed flow preserves Strominger Kähler-like conditions on 2-step nilpotent Lie groups.
The abstract conjectures and verifies a flow on balanced manifolds converging to Kähler metrics.
problem The convergence of pluriclosed flow on balanced manifolds with c1=0. method Analyzes specific cases of compact quotients of Lie groups, verifying the conjecture for invariant metrics.
result The pluriclosed flow on compact balanced manifolds with c1=0 converges to Kähler metrics. We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for stu…
New metrics found on non-Kähler complex manifolds.
problem Finding metrics on non-Kähler complex manifolds.
method Reinterpretation of Bismut Hermitian-Einstein condition and associated holomorphic Courant algebroid.
result Infinitely many non-Kähler manifolds without Bismut Hermitian-Einstein metrics.
Streets and Tian introduced a parabolic flow of pluriclosed metrics. We classify the long time behavior of homogeneous solutions of this flow on closed complex surfaces including minimal Hopf, Inoue, Kodaira, and non-Kahler, properly elliptic surfaces. We also construct expanding soliton solutions to the flow on the un…
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 fun…
Explicitly describes pluriclosed metrics on compact Lie groups.
problem Characterizing pluriclosed metrics on compact Lie groups.
method Explicit description using root systems and invariant structures.
result Explicit formulas for pluriclosed metrics in terms of root systems.
We prove long time existence and convergence results for the pluriclosed flow, which imply geometric and topological classification theorems for generalized Kähler structures. Our approach centers on the reduction of pluriclosed flow to a degenerate parabolic equation for a (1,0)-form, introduced in \cite{ST2}. We ob…
We show that the pluriclosed flow preserves generalized Kähler structures with the extra condition [J+,J−]=0, a condition referred to as "split tangent bundle." Moreover, we show that in this in this case the flow reduces to a nonconvex fully nonlinear parabolic flow of a scalar potential function. We prove a num…
Study on shrinking solitons of generalized Ricci flow.
problem Characterizing shrinking solitons in generalized Ricci flow.
method Analyzing gradient shrinking solitons and pluriclosed solitons on compact manifolds.
result First non-trivial shrinking generalized soliton constructed.
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 at…
We give a classification of compact solitons for the pluriclosed flow on complex surfaces. First, by exploiting results from the Kodaira classification of surfaces, we show that the complex surface underlying a soliton must be Kähler except for the possibility of steady solitons on minimal Hopf surfaces. Then, we const…
The study proves stability of a flow on specific Lie groups.
problem Global stability of the Pluriclosed flow on compact Lie groups.
method Computation of cohomology, verification of flat metrics, and analysis of complex structures.
result Stability of the pluriclosed flow on compact Lie groups of rank two.
We define a parabolic flow of pluriclosed metrics. This flow is of the same family introduced by the authors in \cite{ST}. We study the relationship of the existence of the flow and associated static metrics topological information on the underlying complex manifold. Solutions to the static equation are automatically H…
New flow defined to solve Hull-Strominger system, with estimates and convergence results.
problem Constructing solutions to the Hull-Strominger system of equations.
method Introducing a natural extension of pluriclosed flow and using string algebroids and higher gauge theory.
result Proves global existence and convergence of the flow on special backgrounds.
We study the asymptotic behavior of the pluriclosed flow in the case of left-invariant Hermitian structures on Lie groups. We prove that solutions on 2-step nilpotent Lie groups and on almost-abelian Lie groups converge, after a suitable normalization, to self-similar solutions of the flow. Given that the spaces are so…
Study Hermitian metrics with Bismut connection satisfying Bianchi identity and SKT condition.
problem Characterize Hermitian metrics with Bismut connection satisfying Bianchi identity and SKT condition.
method Analyze SKT condition and Bismut Kähler-like metrics, construct new examples, and study pluriclosed flow.
result Construct new examples of Hermitian manifolds satisfying Bismut Kähler-like condition.
The regularity theory for pluriclosed flow hinges on obtaining Cα regularity for the metric assuming uniform equivalence to a background metric. This estimate was established in \cite{StreetsPCFBI} by an adaptation of ideas from Evans-Krylov, the key input being a sharp differential inequality satisfied by the assoc…
We review some constructions and properties of complex manifolds admitting pluriclosed and balanced metrics. We prove that for a 6-dimensional solvmanifold endowed with an invariant complex structure J having holomorphically trivial canonical bundle the pluriclosed flow has a long time solution for every invariant init…
Classifies simply-connected pluriclosed manifolds with parallel Bismut torsion.
problem Classifying specific types of manifolds with parallel Bismut torsion.
method Complete classification through mathematical analysis.
result Established a splitting theorem for certain manifolds.
We establish Evans-Krylov estimates for certain nonconvex fully nonlinear elliptic and parabolic equations by exploiting partial Legendre transformations. The equations under consideration arise in part from the study of the "pluriclosed flow" introduced by the first author and Tian
Characterizes metrics on Lie groups, proving non-simultaneous existence of balanced and pluriclosed metrics.
problem Existence of balanced and pluriclosed metrics on real semisimple Lie groups.
method Characterization using Vogan diagrams and revisiting complex structure classification.
result Complex manifolds cannot simultaneously admit balanced and pluriclosed metrics.
We prove that compact complex manifolds with admitting metrics with negative Chern curvature operator either admit a ddc-exact positive (1,1) current, or are Kähler with ample canonical bundle. In the case of complex surfaces we obtain a complete classification. The proofs rely on a global existence and convergence …
Characterizes almost abelian Lie algebras with integrable complex structure
problem Classifying almost abelian Lie algebras
method Using presentations consisting of a real number, an element in a vector space, and an endomorphism
result Classifies p-Kähler, p-pluriclosed, Kähler, balanced, pluriclosed, and Gauduchon metrics Characterizes pluriclosed metrics on Oeljeklaus-Toma manifolds.
problem Characterizing existence of pluriclosed metrics on OT manifolds.
method Purely number-theoretical conditions.
result Explicit examples of pluriclosed OT manifolds in arbitrary complex dimension.
Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.
problem Stability of critical points of the generalized Einstein Hilbert action in non-Kähler Calabi-Yau theory.
method Analysis of Bismut Hermitian Einstein manifolds and Bismut flat pluriclosed steady solitons, proving stability conditions.
result All Bismut Hermitian Einstein manifolds are linearly stable, and all Bismut flat pluriclosed steady solitons with positive Ricci curvature are linearly strictly stable.
In this note we observe that on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure the (1,1)-part of the Bismut-Ricci form is seminegative definite. As application we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long…
Book introduces generalized Ricci flow for constructing canonical metrics.
problem Constructing canonical metrics in generalized Riemannian and complex geometry.
method Introduces generalized Ricci flow as a tool for constructing canonical metrics.
result Global convergence results and applications to complex geometry.
Develops a new framework for generalized Ricci flow on Lie groups.
problem Global existence and geometric properties of Ricci flow on Lie groups.
method Inspired by Lauret's bracket flow, studies generalized Ricci flow on discrete quotients of Lie groups.
result Establishes global existence on solvmanifolds in arbitrary dimensions.
Balanced metrics found on Lie groups and their quotients.
problem Existence of balanced metrics on Lie groups and quotients.
method Proved existence of invariant complex structures and Hermitian balanced metrics on Lie groups and quotients.
result Existence of balanced metrics on Lie groups and quotients, and no pluriclosed metrics.
Study on properties of Oeljeklaus-Toma manifolds, including cohomology and metrics.
problem Characterizing and understanding the metric and cohomological properties of Oeljeklaus-Toma manifolds.
method Analysis of double complex of differential forms, Bott-Chern cohomology, and explicit formulas for Dolbeault cohomology.
result Proved that Oeljeklaus-Toma manifolds do not admit certain types of metrics and provided explicit formulas for their Dolbeault cohomology.
Characterizes complex structures on specific Lie groups.
problem Identifying Lie groups with left-invariant complex structures.
method Analyzing Lie algebras and their corresponding Lie groups, considering different nilpotency levels.
result Conditions for the existence of left-invariant complex structures and pluriclosed metrics on 2-step nilpotent Lie groups.
Study of Hermitian structures on toric suspensions of balanced manifolds.
problem Exploring Hermitian structures on specific types of manifolds.
method Analysis of toric suspensions of Calabi-Yau and hyperkähler manifolds under holomorphic automorphisms.
result Suspensions of hyperkähler manifolds do not admit certain Hermitian metrics.
Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in arXiv:1802.00170. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first example…