New flow preserves almost Hermitian metrics for manifold study.
problem Curvature flow for almost Hermitian manifolds.
method Introducing a new curvature flow matching Ricci flow and preserving almost Hermitian condition.
result Ricci flow can be used to study almost Hermitian manifolds.
Unified flow approach to curvature problem on specific manifolds.
problem Prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree.
method Unified flow approach with conditions on curvature function f. result Flow converges to a conformal Hermitian metric with specified curvature.
Study shows curvature of Hermitian-Yang-Mills flow is uniformly bounded.
problem Curvature estimate of Hermitian-Yang-Mills flow on holomorphic vector bundles.
method Uniform curvature bound demonstrated in a simple case.
result Curvature of evolved Hermitian metric is uniformly bounded.
Study Hermitian curvature flow on complex surfaces, finding singularities and limits.
problem Characterize and analyze Hermitian curvature flow on complex surfaces.
method Case-by-case analysis of flow on each complex model geometry.
result First example of a compact complex non-Kähler manifold with finite time singularity.
Finite extinction time for Hermitian curvature flow on homogeneous spaces.
problem Finite extinction time of Hermitian curvature flow on compact homogeneous spaces.
method Proved finite extinction time and analyzed flow behavior.
result Finite extinction time T>0 for the flow. Study Bismut-Griffiths-positivity in non-Kähler manifolds under Hermitian curvature flows.
problem Investigate positivity of Bismut curvature in non-Kähler manifolds.
method Analyze Bismut-Griffiths-positivity under Hermitian curvature flows.
result Identify HCFs that do not preserve Bismut-Griffiths-positivity.
Compact Hermitian manifolds with quasi-negative curvature have ample canonical line bundles.
problem Determining conditions for ample canonical line bundles in Hermitian manifolds.
method Hermitian curvature flow with specific curvature conditions.
result Canonical line bundle is ample under given curvature conditions.
Study the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
problem Understanding the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
method Monotonicity of eigenvalues of mean curvature, convergence to geometric invariants.
result Eigenvalues of mean curvature converge to geometric invariants in the Gauduchon case.
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.
A new flow of Hermitian metrics reduces to a scalar equation and preserves special structures.
problem Evolution of Hermitian metrics to preserve special structures.
method Introducing a scalar Calabi-type flow that depends on a background metric.
result The flow has a unique short-time solution and stability when the background metric is Kaehler-Einstein with nonpositive scalar curvature.
Study of curvature flow on complex Lie groups, leading to soliton convergence.
problem Characterizing long-time behavior of curvature flow on complex 2-step nilpotent Lie groups.
method Analyzing left-invariant metrics and using Cheeger-Gromov topology.
result Normalized solutions converge to a non-flat algebraic soliton.
The paper proves the Hermitian Curvature Flow preserves various curvature conditions.
problem Proving the preservation of curvature conditions under the Hermitian Curvature Flow.
method Constructing convex sets of curvature operators invariant under the HCF and varying parameters to prove preservation.
result The Hermitian Curvature Flow preserves Griffiths positivity, Dual-Nakano positivity, and positivity of holomorphic orthogonal bisectional curvature.
Study on Hermitian curvature flow on special linear groups, disproving a conjecture and finding non-algebraic solitons.
problem Disproving a conjecture about the stability of canonical metrics on special linear groups.
method Investigation of invariant solutions to the Positive Hermitian Curvature Flow on complex Lie groups.
result Discovered non-algebraic solitons on special linear groups, contradicting Ustinovskiy's conjecture.
The Almost Hermitian Curvature flow was introduced by Streets and Tian in order to study almost hermitian structures, with a particular interest in symplectic structures. This flow is given by a diffusion-reaction equation. Hence it is natural to ask the following: which almost hermitian structures are dynamically stab…
Regularities and stability shown for a specific type of complex parallelizable manifolds.
problem Stability and regularity of Chern-flat metrics on complex parallelizable manifolds.
method Study of Hermitian metrics governed by the second Chern-Ricci form on compact complex manifolds.
result Chern-flat metrics are dynamically stable on compact complex parallelizable manifolds.
New bounds for geometric flows of Hermitian metrics established.
problem Regularity of geometric flows of Hermitian metrics.
method Establishing a C1 a priori bound for smooth curves of Hermitian metrics. result New regularity result for Hermitian curvature flows, including the second Chern-Ricci flow.
In this paper we study a particular version of the Hermitian curvature flow (HCF) over a compact complex Hermitian manifold (M,g,J). We prove that if the initial metric has Griffiths positive (non-negative) Chern curvature Ω, then this property is preserved along the flow. On a manifold with Griffiths non-negative …
Flow smooths Chern-Ricci-flat metrics on Hermitian manifolds.
problem Smooth Chern-Ricci-flat metrics on Hermitian manifolds.
method An analogue of the Calabi flow for compact Hermitian manifolds with vanishing first Bott-Chern class.
result The flow converges to the unique Chern-Ricci-flat metric under certain conditions.
The paper examines the stability of a specific flow on complex manifolds.
problem Stability of line bundle mean curvature flow on complex manifolds.
method Analyzes the convergence of the line bundle mean curvature flow to a deformed Hermitian-Yang-Mills metric.
result The flow converges exponentially to the deformed Hermitian-Yang-Mills metric in the C∞ sense. We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler-Lagrange equation for this functional is an elliptic equation for Hermitian metrics. Solutions to this equation are related to Kähler-Einstein metrics, and are automatically Kähler-Einstein under certain conditions. Give…
Study of curvature flow on specific Lie groups, leading to soliton solutions.
problem Curvature flow on 2-step nilpotent Lie groups with complex structures.
method Left-invariant metrics and complex structures on Lie groups, convergence analysis.
result Existence and convergence of flow to soliton solutions.
The paper preserves positivity along a flow over projective bundles.
problem Preserving positivity in geometric flows over projective bundles.
method Introducing a flow over projective bundles and proving semipositivity preservation under certain conditions.
result Semipositivity of curvature is preserved along the flow under specific conditions.
We introduce a parabolic flow of almost Kahler structures, providing an approach to constructing canonical geometric structures on symplectic manifolds. We exhibit this flow as one of a family of parabolic flows of almost Hermitian structures, generalizing our previous work on parabolic flows of Hermitian metrics. We e…
Study on prescribing curvature on specific manifolds with negative Gauduchon degree.
problem Prescribing Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree.
method Analysis of geometric flow convergence to obtain existence results.
result Existence results for curvature functions that are nonzero and nonpositive, and sign-changing cases.
In recent years, Streets and Tian introduced a series of curvature flows to study non-Kähler geometry. In this paper, we study how to construct second order curvature flows in a uniform way, under some natural assumptions which holds in Streets and Tian's works. As a result, by classifying the lower order tensors, we c…
On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (Riemannain real vector bundle) with an arbitrary metric connection over a compact Hermitian manif…
Study of HCF on Lie groups leads to static metrics.
problem Investigating Hermitian curvature flow on Lie groups.
method Ricci-flow type equation and convergence analysis.
result Existence and convergence of solutions to HCF.
The paper proves properties of metrics and self-shrinkers related to Hermitian-Yang-Mills and J-equations.
problem Properties of metrics and self-shrinkers related to Hermitian-Yang-Mills and J-equations.
method Proved parallel curvatures of deformed Hermitian-Yang-Mills metrics and properties of self-shrinkers over Cn. result Proved that curvatures of deformed Hermitian-Yang-Mills metrics are parallel with respect to the background metric.
The paper discusses pseudo-harmonic maps and their heat flow from pseudo-Hermitian to Riemannian manifolds.
problem Existence and uniqueness of pseudo-harmonic maps.
method Heat flow approach for pseudo-harmonic maps from pseudo-Hermitian to Riemannian manifolds.
result Existence and uniqueness of pseudo-harmonic maps under certain curvature conditions.
The paper studies a flow on complex homogeneous manifolds, proving invariance and constructing HCF-Einstein metrics.
problem Understanding the behavior of Hermitian curvature flow on complex homogeneous manifolds.
method Investigation of a version of the Hermitian curvature flow on complex homogeneous manifolds, focusing on induced metrics and Lie algebra computations.
result Construction of HCF-Einstein metrics on G-homogeneous manifolds and investigation of blow-up behavior for nilpotent or solvable Lie groups. Survey on metrics on non-Kähler complex manifolds.
problem Existence and properties of Hermitian metrics.
method Analytic study of Chern connection and related flows.
result Generalizations of Kähler-Einstein condition.
The paper studies a flow on complex Lie groups, showing convergence to solitons.
problem The study of curvature flows on complex Lie groups.
method Positive Hermitian curvature flow on left-invariant metrics.
result The flow converges to solitons in both nilpotent and almost-abelian cases.
The paper connects bundle curvature to random zero currents.
problem Understanding the relationship between bundle curvature and random zero currents.
method Heat flow on Hermitian line bundles over Riemannian manifolds.
result Random zero currents connect bundle curvature to ground state zero current.
The paper studies singularities in a complex flow related to mean curvature.
problem Investigating singularities in a complex flow related to mean curvature.
method Constructing two distinct examples of singularities using the line bundle mean curvature flow.
result Found a finite time singularity, ruling out long time existence of the flow.
Survey of recent geometric flows in complex geometry.
problem Preserving conformally balanced property of Hermitian metrics.
method Anomaly flow, a flow of (2,2)-forms on a 3-fold. result Anomaly flow is a higher order extension of the Ricci flow.
We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way, by considering a generic flow under just a few natural conditions on the broad class of almost-hermitian structures. As a main tool, we use an ODE system defined on the variety …
Study expanding solitons on complex Lie groups with specific algebraic structures.
problem Investigate expanding solitons on complex Lie groups with left-invariant metrics.
method Analyze the algebraic structure of complex Lie groups and their decompositions.
result Show that the Lie algebras of complex Lie groups decompose into semidirect products.
Study on Hermitian manifolds with curvature, finding geometric properties.
problem Understanding the structure of Hermitian manifolds with semipositive Griffiths curvature.
method Combining HCF, torsion-twisted connection properties, and geometric observations.
result Null spaces of the Chern-Ricci form generate a holomorphic, integrable distribution.
The paper proves a condition for a metric on a vector bundle to be Griffiths positive.
problem Proving that a hermitian metric on a vector bundle is Griffiths positive.
method Defined a flow on the projective bundle and argued for its convergence.
result The relative Kähler-Ricci flow on the projective bundle converges.
Study on line bundle flow on Kähler surfaces converging to a singular solution.
problem Analyzing the mean curvature flow on Kähler surfaces.
method Investigates the flow under hypercritical phase and semipositivity conditions.
result The flow converges to a singular solution away from curves of negative self-intersection.
Anomaly flow uses superstring theory to maintain metric properties.
problem Maintaining metric properties in superstring theory.
method Green-Schwarz anomaly cancellation mechanism, preserving conformally balanced condition.
result Explicitly infers Hermitian metric flow from its (m−1)-th power. Study Gauduchon connections on Kähler-like metrics in 6D solvmanifolds.
problem Investigate Hermitian metrics with Gauduchon connections having symmetries similar to Levi-Civita and Chern connections.
method Analyze 6-dimensional solvmanifolds with invariant complex structures and trivial canonical bundles. result Evidence supports two conjectures about Kähler-like conditions for different Gauduchon connections.
The CR Yamabe flow converges exponentially to a contact form with flat curvature.
problem Analyzing the CR Yamabe flow with zero invariant.
method Used CR Poincaré inequality and Gagliardo-Nirenberg type interpolation inequality.
result The flow converges exponentially to a contact form with flat pseudo-Hermitian scalar curvature.
The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flo…
We produce solutions to the Kähler-Ricci flow emerging from complete initial metrics g0 which are C0 Hermitian limits of Kähler metrics. Of particular interest is when g0 is Kähler with unbounded curvature. We provide such solutions for a wide class of U(n)-invariant Kähler metrics g0 on n dimensional c…
In this paper, we generalize Medos-Wang's arguments and results on the mean curvature flow deformations of symplectomorphisms of $\CP^n$ in \cite{MeWa} to complex Grassmann manifold $G(n, n+m;\C)$ and compact totally geodesic Kähler-Einstein submanifolds of $G(n, 2n;\C)$ such as irreducible Hermitian symmetric spaces $…
We show that (a) any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in Cm with the Euclidean metric is flat; (b) any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in Cm with the pseudo-Euclidean metric is flat if the H…
Defines a volume functional for Hermitian connections on manifolds, proving its properties and connections.
problem Defining and analyzing volume functionals for Hermitian connections on manifolds.
method Introduces the line bundle mean curvature flow and relates it to deformed connections and special submanifolds.
result Proves the mirror equality for mSpin(7)-dDT connections and deduces their properties.