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.
Hermitian metrics with zero second Chern Ricci curvature are rigid and exist on specific manifolds.
problem Characterizing Hermitian metrics with vanishing second Chern Ricci curvature.
method Analyzing the rigidity of the second Chern Ricci curvature on compact complex manifolds.
result Characterization of second Chern Ricci-flat Hermitian metrics and non-existence results.
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…
In this work, we first establish short time existence and Shi's type estimate of second Ricci flow on complete noncompact Hermitian manifolds. As an application, we use the second Ricci flow to discuss the existence of Kaehler-Einstein metric on complete noncompact Hermitian manifolds.
Study proves Kählerness criteria for Hermitian surfaces under specific curvature conditions.
problem Determining when Hermitian surfaces are Kählener.
method Used explicit identities linking Strominger-Bismut Ricci curvatures to torsion, and Chern number identities.
result Proves several Kählerness criteria for compact Hermitian surfaces.
Study deforms Hermitian metrics with positive curvature.
problem Deforming Hermitian metrics with positive curvature.
method Adapted conformal perturbation method to Hermitian setting.
result Hermitian metrics with quasi-positive curvature can be deformed to positive curvature.
Uniform estimates for complex Monge-Ampère equations on hermitian varieties.
problem Uniform boundedness of Chern-Ricci flat potentials in conifold transitions.
method Proving uniform a priori estimates for degenerate complex Monge-Ampère equations.
result Generalization of a theorem to hermitian contexts.
Survey on Chern-Ricci flow for complex manifolds.
problem Understanding and solving open problems in Chern-Ricci flow.
method Parabolic flow of Hermitian metrics on complex manifolds.
result Open problems and new directions in Chern-Ricci flow highlighted.
The study examines deformations of Ricci-flat ALF spaces, showing they must be Hermitian.
problem Classifying and understanding deformations of Ricci-flat ALF spaces.
method Assuming suitable fall-off conditions, the authors show that deformations must be Hermitian and carry a non-trivial Killing vector field.
result The new Ricci-flat metric must belong to the family of previously classified metrics under mild additional hypotheses.
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.
Proves unique ALE instanton with toric Hermitian structure.
problem Classify Ricci flat ALE instantons with toric Hermitian non-Kähler structure.
method Direct global analysis of Tod form in Weyl-Papapetrou coordinates, avoiding toric Kähler geometry.
result Eguchi-Hanson instanton is the only smooth, Ricci flat, ALE instanton with toric Hermitian non-Kähler structure.
Article establishes criteria for multiplier Hermitian-Einstein metrics on KSM-manifolds.
problem Existence of multiplier Hermitian-Einstein metrics on Fano manifolds.
method Criterion based on KSM-data and continuous paths connecting solitons.
result Explicit example of a KSM-manifold with a family of multiplier Hermitian-Einstein metrics.
The study examines geometric properties of complex Hermitian manifolds and their holonomy groups.
problem Understanding the geometric properties and restrictions of Hermitian manifolds and their holonomy groups.
method Analyzing the representation of restricted holonomy groups and their geometric consequences.
result Established criteria for when a Hermitian manifold is Kähler or projective based on its holonomy group.
New metrics found on non-Kähler Calabi-Yau manifolds.
problem Finding Levi-Civita Ricci-flat metrics on non-Kähler Calabi-Yau manifolds.
method Constructing new metrics on specific types of non-Kähler Calabi-Yau manifolds.
result Examples of Levi-Civita Ricci-flat metrics on various non-Kähler Calabi-Yau manifolds.
We prove that a general complex Monge-Ampère flow on a Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti-Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generaliza…
Study fully nonlinear equations on Hermitian manifolds to find metrics with specific curvature.
problem Finding conformal Hermitian metrics with prescribed curvature functions.
method Blow-up argument and partial uniform ellipticity.
result Our assumptions are almost sharp, with some geometric function theory obstructions.
The paper explores properties of Gauduchon curvature in Hermitian manifolds.
problem Investigating properties of Gauduchon curvature in Hermitian manifolds.
method Analyzing the Ricci curvature of Gauduchon connections and proving existence of metrics.
result Monotonicity theorem for Gauduchon holomorphic sectional curvature.
New metrics found on non-Kähler Calabi-Yau manifolds.
problem Constructing Ricci-flat metrics on non-Kähler Calabi-Yau manifolds.
method Using t-Gauduchon metrics on principal torus bundles over rational homogeneous varieties. result Examples of new metrics on non-Kähler Calabi-Yau manifolds.
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.
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.
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.
In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the (1,1)- component of the curvature 2-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Her…
Gradient estimates derived for a specific equation on pseudo-Hermitian manifolds.
problem Deriving gradient estimates for solutions of a specific equation on pseudo-Hermitian manifolds.
method Derives gradient estimates for positive solutions of the equation Δbu+aup+1=0 on pseudo-Hermitian manifolds. result Gradient estimates obtained for the positive solutions of the equation.
The paper classifies 4D Ricci-flat ALE manifolds and their properties.
problem Characterizing 4D Ricci-flat ALE manifolds and their properties.
method Analyzing the correspondence between ALE gravitational instantons and Kähler orbifolds, and proving properties of these structures.
result There is a one-to-one correspondence between Hermitian non-Kähler ALE gravitational instantons and Bach-flat Kähler orbifolds in 2D.
In this work, we show that along a particular choice of Hermitian curvature flow, the non-positivity of Chern-Ricci curvature will be preserved if the initial metric has non-positive bisectional curvature. As an application, we show that the canonical line bundle of a compact Hermitian manifold with nonpositive bisecti…
We introduce transverse Chern-Ricci flow for transversely Hermitian foliations, which is analogous to the Chern-Ricci flow. We show that when F is homologically orientable and the basic first Bott-Chern class is zero, starting at any transversely Hermitian metric the flow exists for all time and as $t\right…
Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kaehler-Weyl structure; this structure is locally conformally Kaehler if and only if the alternating Ricci tensor vanishes. The alternating Ricci tensor takes values in a certain representation space. In this paper, we show that any algebrai…
Survey of recent progress in gravitational instantons
problem Classification of hyperkähler and Hermitian gravitational instantons
method Survey of recent progress in gravitational instantons
result Survey of recent progress in gravitational instantons
Study finds criteria for surfaces with specific curvature properties.
problem Understanding Kählerian or projective structures on surfaces with non-positive curvature.
method Established a criterion for compact Hermitian surfaces with non-positive second Chern-Ricci curvature.
result Found conditions for Kählerian or projective structures on surfaces with non-positive curvature.
Study classifies gravitational instantons based on their asymptotic geometry.
problem Classifying gravitational instantons based on their asymptotic properties.
method Investigation of asymptotic geometry of Hermitian non-Kähler Ricci-flat metrics.
result All Hermitian non-Kähler gravitational instantons can be compactified to log del Pezzo surfaces.
This paper is concerned with Chern-Ricci flow evolution of left-invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger-Gromov) sense to a Chern-Ricci …
We show that on a smooth Hermitian minimal model of general type the Chern-Ricci flow converges to a closed positive current on M. Moreover, the flow converges smoothly to a Kahler-Einstein metric on compact sets away from the null locus of K_M. This generalizes work of Tsuji and Tian-Zhang to Hermitian manifolds, prov…
Vanishing theorems show holomorphic tensor fields on certain Kähler manifolds are trivial.
problem Understanding properties of holomorphic tensor fields on Kähler manifolds.
method Established vanishing theorems for uniformly rational connected (RC) k-positive Hermitian holomorphic vector bundles. result Holomorphic tangent bundles of Kähler manifolds with positive k-Ricci curvature are uniformly RC k-positive. 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.
We consider the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci form. This is an evolution equation first studied by M. Gill, and coincides with the Kahler-Ricci flow if the initial metric is Kahler. We find the maximal existence time for the flow in terms of the initial data. We invest…
In this note, using Calabi's method, we construct rotationally symmetric Kahler-Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao an…
The paper explores mixed curvature for Hermitian manifolds and its implications.
problem Investigating the properties of mixed curvature for Hermitian manifolds.
method Analyzing convex combinations of first Chern Ricci curvature and holomorphic sectional curvature.
result Compact Hermitian surfaces with constant mixed curvature are Kähler unless specific conditions are met.
Uniform estimates lead to Gromov-Hausdorff limits for Hermitian minimal models.
problem Uniform diameter and volume estimates for Chern-Ricci flow on Hermitian minimal models.
method Uniform diameter and volume estimates, local Kähler assumption, Perelman's reduced length, almost monotonicity formula for reduced volume.
result Gromov-Hausdorff convergence of the Chern-Ricci flow on Hermitian minimal models.
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.
Gravitational instantons are non-Kähler, providing a counterexample to Euclidean Black Hole Uniqueness.
problem Classical Euclidean Black Hole Uniqueness conjecture
method Analyzing the Chen-Teo gravitational instanton
result The Chen-Teo instanton is Hermitian and non-Kähler
Defines and studies solutions to complex equations on Hermitian manifolds.
problem Solving complex equations on Hermitian manifolds.
method Extending recent theories, defines and studies pluripotential solutions to degenerate parabolic complex Monge-Ampère equations.
result Establishes existence and uniqueness of weak Chern-Ricci flow on complex compact varieties with log terminal singularities.
The aim of this paper is to classify bi-Hermitian compact surfaces (M,g) whose Ricci tensor ρ satisfies the relation ∇Xρ(X,X)=31Xτg(X,X).
In this paper we give new examples of QCH Kahler surfaces whose opposite almost Hermitian strucure is Hermitian and not locally conformally Kahler. In this way we give also a large class of examples of Hermitian surfaces with J-invariant Ricci tensor which are not l.c.k.
Estimates eigenvalue for Hermitian manifolds using curvature.
problem Estimating the first eigenvalue of Hermitian manifolds.
method Using holomorphic Ricci and sectional curvatures.
result Established estimates for the first eigenvalue.
The paper explores Ricci forms on noncompact complex manifolds, including Kähler-Einstein and canonical metrics.
problem Existence of specific metrics on noncompact complex manifolds with prescribed Ricci curvature.
method Analyzes geometric problems on noncompact complex manifolds using Ricci curvature.
result Improves the main theorem in Cheng-Yau [4] and constructs Hesse-Einstein metrics.
We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative c…
In this paper, we will give a horizontal gradient estimate of positive solutions of Δbu=−λu on complete noncompact pseudo-Hermitian manifolds. As a consequence, we recapture the Liouville theorem of positive pseudo-harmonic functions on Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature.
Study bi-Hermitian metrics on complex surfaces and solve geometric PDEs.
problem Construct canonical metrics on complex surfaces with split tangent bundle.
method Introduced new fully non-linear geometric PDEs and established smooth solutions.
result Solved the prescribed Bismut Ricci problem on complex surfaces.